I'm sure a non-insignificant group do move to one spouse's hometown, since I can assure you, having your kids' grandparents close by can be life-changing (especially for modern, dual-income families).
I'm seeing a lot less of that among people my age who are having kids. Most stay wherever they've setup their lives but move to a larger house or something similar.
I see the same, but I think the consequences hinted at by the GP are very present. I know quite a few parents, some with their parents nearby, others without. The ones with their parents nearby universally have more of an ability to do things without their kids (either regularly, or as a special-case).
And I know several parents who tried the "go it alone" method for a while, but eventually decided to move back to one of their hometowns, specifically to be closer to their parents for reasons related to their kids. Most of it seems to be a mix of wanting nearby, reliable childcare, as well as just wanting their kids to grow up with regular contact with their grandparents. (I've also seen the phenomenon where the grandparents, after retirement, end up moving to the parents' location, but often that requires the grandparents to move to a higher cost-of-living area, so I expect that happens less often.)
I don't plan to have kids, so this is certainly a grain-of-salt-worthy opinion, but it feels to me that parent-marriages would be on average healthier if the parents had easier childcare options so they can more often do "date night" type things. Certainly nearby grandparents is not the only way to achieve that, but it is one of the simplest options, assuming grandparents who are happy to be involved to that degree.