Band Steering will cause the routers and access points to kick you off 5Ghz faster, if 2.4Ghz would give you stronger signal, and vice-versa.
Router Steering will cause the routers and access points to kick you off the one you are attached to more quickly, and thus cause you to re-associate with the network on a router or access point that is closer to you and would give you a stronger signal.
I can't speak for any other vendors, but that's the kind of thing I think you're looking for.
I tried to find the Apple whitepaper I read about this exact topic a couple years ago, but couldn't find it. Hopefully someone with more precise insight can chime in.
One practical solution may be to reduce the signal strength of each of your access points so the threshold for jumping APs is hit more easily.
If you name them the same, they should auto-switch more aggressively, but probably still not optimally.
For that communication between the routers is needed and each will kick you over to the other when they agree it’s necessary
Recommended reading: https://support.apple.com/en-gb/HT202628
As for mesh networking, nice to have in the suburbs, though never as good as a fully-managed, wired-AP solution, especially in Manhattan high-rises.