You need two black wires - line in and load out, and a white wire - neutral. If you don't have that 2nd black you've probably got a "power through fixture" circuit and the switch won't work there.

if by "replace the nonsmart switches" you mean change those to smart switches that will have no effect. If you replaced the old switch there's a small chance the new one is broken too, but more...

If that doesn't work, check your router specs, logs, and status. It's firewall may be blocking outgoing NTP requests from the tplink devices.

Not safely, you need all 3 wires. If they're not available the box needs to be re-wired.

It probably won't work there. You need a neutral wire, line in, and load going to the fixture. You might have power through fixture type wiring which is a problem for smart switches.

That's the response TP Link devices use when you try to set rules on a device that can't reach the internet/network time servers.