Concurrent opinion here. I have a lot of things to say about this, but it boils down to, "I think this guy may be expecting to be the first priority of someone who is his third priority." And he's chosen some very legit priorities--- his kids, and lucrative work with job requirements he's comfortable with! These are good choices! Well done, LW, you seem to have your priorities in order!
But. Either, as you said, he needs to improve his selection of partners, or he needs to rethink his definition of "serious". Because as you say there are definitely women who would date this guy on a level they consider serious. But finding someone who is okay with, in effect, being his third priority almost certainly means someone who will return the favor in some way. In particular, if he's looking for people who want to cohabitate and/or share community property, his overall package deal is... definitely narrowing the field. "We move constantly, our primary permanent social circle seems to have ended up revolving around my husband's first wife, her new spouse, and her offspring not all of whom are his but he's not interested in having kids with me, and I can't keep a job or friends of my own because we're always moving" is... its own advice column letter, to say the least. And... is he expecting a partner to stay over at the first wife's house with him when he goes to see the kids? More generally, how much is he expecting a partner to spend time, not just with his offspring, but with his first wife and new husband? It's one thing to accept that kids are a package deal with their parents and that you're going to need to at least attempt a modicum of civility with the other parent of your partner's kids, but this is potentially a LOT of socializing with the same two other adults, in their home. That's a three-for-one deal that may not be a bargain for some people.
I'll admit that my own brain catches on "you seem to be suggesting that you want someone who doesn't want to have kids but wants to spend time with yours" which is going to narrow the field all on its own--- not that everyone who wants kids feels the need to have them be biological, but you're having to want these specific kids while not adopting/birthing any yourself or even having primary custody of the ones you are only-sort-of-getting to parent. But the "I move so frequently for work that there's no point in owning a home even though I make good money" also sounds like someone who... either needs a long-term partner who's equally mobile in some way or who needs to adjust to LDR's, because however good it is for LW's career, it's probably not good for a partner's work and other financial and psychosocial prospects unless they're in a really flexible or high-demand profession or something, and that's not even getting into whatever social support networks the partner might be having to uproot--- yes, there are people who can do this! But LW is just as-is making some unusual choices and making them work, and it's going to take someone who's any of a number of different right kinds of unique in their own right to mesh with what he's got working rather than wreck it.
Which is a really long way of saying that I agree that he needs better partner-selection processes. Or... reconsider some of what he wants in a relationship, maybe? Like, someone who wholly supports him spending time with the kids but isn't quite as involved as he'd like? Or someone who gets along with the kids and ex-and-new-spouse and is, like, willing to co-purchase a "home base"--- maybe in the same area as ex-wife!--- and not move all over the place with him so she can keep her job and they'll see each other when they can, and also the kids can all come and stay at their place sometimes? (Minus the "kids and ex-wife and spouse" part, my parents did this for a while after I went to college--- Mom had a job she liked, so Dad moved for his job and she stayed and they saw each other when they both had vacation.)
Also, I have to wonder how he would be with the shoe being on the other foot--- he talks about "not looking to start a second family at the expense of my first" and also about women "expect[ing] him to jump through hoops for their families". Is he dating other people with children? How would he be with someone he was serious with who had an equivalent relationship with her ex-husband? Would he sleep over at his partner's first husband's house with her? What role does he see for himself in blending any kids his new partner might already have into the bigger family? Because that "not looking to start a second family"... okay, I'm glad he's putting the kids he's already got first! But... the pool of people his age who like being around kids just enough to want to spend time with his kids, at his ex-wife's place, but who don't either already have kids or want more involvement in the parenting of kids than they may get in this scenario gets smaller and smaller. I'm not saying they don't exist! It's just--- it's sort of like in all the advice columns where one half of a couple writes in about how to compromise on the "kids/no kids" issue and someone says, "You can't have half a baby"? This is like the flip side of that, because LW's partner is likely getting half a parenting experience--- and it's going to tricky finding someone who wants exactly that half. (And then we get back to the geographic instability and... yeah.)
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But. Either, as you said, he needs to improve his selection of partners, or he needs to rethink his definition of "serious". Because as you say there are definitely women who would date this guy on a level they consider serious. But finding someone who is okay with, in effect, being his third priority almost certainly means someone who will return the favor in some way. In particular, if he's looking for people who want to cohabitate and/or share community property, his overall package deal is... definitely narrowing the field. "We move constantly, our primary permanent social circle seems to have ended up revolving around my husband's first wife, her new spouse, and her offspring not all of whom are his but he's not interested in having kids with me, and I can't keep a job or friends of my own because we're always moving" is... its own advice column letter, to say the least. And... is he expecting a partner to stay over at the first wife's house with him when he goes to see the kids? More generally, how much is he expecting a partner to spend time, not just with his offspring, but with his first wife and new husband? It's one thing to accept that kids are a package deal with their parents and that you're going to need to at least attempt a modicum of civility with the other parent of your partner's kids, but this is potentially a LOT of socializing with the same two other adults, in their home. That's a three-for-one deal that may not be a bargain for some people.
I'll admit that my own brain catches on "you seem to be suggesting that you want someone who doesn't want to have kids but wants to spend time with yours" which is going to narrow the field all on its own--- not that everyone who wants kids feels the need to have them be biological, but you're having to want these specific kids while not adopting/birthing any yourself or even having primary custody of the ones you are only-sort-of-getting to parent. But the "I move so frequently for work that there's no point in owning a home even though I make good money" also sounds like someone who... either needs a long-term partner who's equally mobile in some way or who needs to adjust to LDR's, because however good it is for LW's career, it's probably not good for a partner's work and other financial and psychosocial prospects unless they're in a really flexible or high-demand profession or something, and that's not even getting into whatever social support networks the partner might be having to uproot--- yes, there are people who can do this! But LW is just as-is making some unusual choices and making them work, and it's going to take someone who's any of a number of different right kinds of unique in their own right to mesh with what he's got working rather than wreck it.
Which is a really long way of saying that I agree that he needs better partner-selection processes. Or... reconsider some of what he wants in a relationship, maybe? Like, someone who wholly supports him spending time with the kids but isn't quite as involved as he'd like? Or someone who gets along with the kids and ex-and-new-spouse and is, like, willing to co-purchase a "home base"--- maybe in the same area as ex-wife!--- and not move all over the place with him so she can keep her job and they'll see each other when they can, and also the kids can all come and stay at their place sometimes? (Minus the "kids and ex-wife and spouse" part, my parents did this for a while after I went to college--- Mom had a job she liked, so Dad moved for his job and she stayed and they saw each other when they both had vacation.)
Also, I have to wonder how he would be with the shoe being on the other foot--- he talks about "not looking to start a second family at the expense of my first" and also about women "expect[ing] him to jump through hoops for their families". Is he dating other people with children? How would he be with someone he was serious with who had an equivalent relationship with her ex-husband? Would he sleep over at his partner's first husband's house with her? What role does he see for himself in blending any kids his new partner might already have into the bigger family? Because that "not looking to start a second family"... okay, I'm glad he's putting the kids he's already got first! But... the pool of people his age who like being around kids just enough to want to spend time with his kids, at his ex-wife's place, but who don't either already have kids or want more involvement in the parenting of kids than they may get in this scenario gets smaller and smaller. I'm not saying they don't exist! It's just--- it's sort of like in all the advice columns where one half of a couple writes in about how to compromise on the "kids/no kids" issue and someone says, "You can't have half a baby"? This is like the flip side of that, because LW's partner is likely getting half a parenting experience--- and it's going to tricky finding someone who wants exactly that half. (And then we get back to the geographic instability and... yeah.)