I have 2 routers and 3 points, all nest Wi-Fi.
1 router connected to wan as NAT.
All other as bridge of course.
2nd router is not connected to wan, (so wireless only) it has a switch connected to it via its Lan port. All the downstream get no ip from dhcp, they set to self assigned. Tried bypassing switch, different cables. It did work well and fast after I installed it, but now stopped working. Let me know if anyone knows.
No, unmanaged is what you want. I got a GS108 from Netgear, it works well and have done so for years now.
Have you tried a power cycle of Nest Wifi and the switch? If that makes no difference, try a factory reset and set it up from scratch?
This should "just work", so I have no good eplaination why it doesn't in your case.