ESPHome devices disconnect from Home Assistant until the integration is reloaded

### The problem

Since updating to Home Assistant 2021.12.7, I have several esphome nodes (esp8266 boards with a couple of older versions of esphome, 1.14.3 and 1.16.0) that will randomly show as unavailable in HA, and the only way to make them available again is to reload the integration in HA (Integrations -> ESPHome -> select the particular node, and Reload).

These few boards in particular are on the far edge of my wifi range. But I've never had them not reconnect back to HA before, and I've never had to reload the integration to get them operable again. I've made no other recent changes to anything except updating to HA 2021.12.7 (from 2021.11.x).

I can still access the web portal for each node and see the logs that way, even when they show "unavailable" in HA.

This seems to be something new in HA 2021.12.x as I never had issues before then. These nodes have been running reliably for almost a year now.

Is this a HA issue? Would this be corrected by updating the nodes to the most recent ESPHome version? Any advice would be appreciated!

### Which version of ESPHome has the issue?

2021.12.3

### What type of installation are you using?

pip

### Which version of Home Assistant has the issue?

2021.12.7

### What platform are you using?

ESP8266

### Board

Wemos D1 mini

### Component causing the issue

_No response_

### Example YAML snippet

_No response_

### Anything in the logs that might be useful for us?

```txt
Noticed this after connecting via the web portal and before reloading the HA integration:

[E][api:084]: No client connected to API. Rebooting...
[I][app:132]: Forcing a reboot...
[D][api.connection:604]: Client 'Home Assistant 2021.12.7 (192.168.1.4)' connected successfully!
[D][time:040]: Synchronized time: Tue Jan 11 11:24:03 2022
```

### Additional information

_No response_