I do have a couple of cameras that are 2.4GHz only, but they normally never get disconnected. It's true that the Nest was dual band, though.
By the way, how do you know which firmware is installed? I can't see it anywhere...
Recent firmware versions support channels 12 and 13. If you were on a firmware version prior to this it just wouldn't see your network at all if it was on channel 12 or 13.
The Evotouch is 2.4Ghz 802.11b/g/n only so having the same SSID on 2G and 5G won't cause you an issue as it won't even see the 5G network.
One possibility is that your 2.4Ghz WiFi is not working well (interference from neighbours or other devices etc) and you don't realise it because the vast majority of devices these days are dual band 2G and 5G and will tend to use the 5G band. The Evotouch is probably one of the few perhaps only WiFi device in your house that is 2G only therefore will suffer if you have interference issues on 2G.
Our baby monitor used to wipe out 2.4Ghz WiFi - basically unusable for browsing and other normal use but 5G perfectly usable. Surprisingly the Evotouch managed to hang on and stay connected on 2G even when laptops were struggling as its bandwidth requirements are probably miniscule.
I would lock a device like a laptop on 2G (by using a unique 2.4Ghz only SSID name) and do some speed testing of your 2.4Ghz signal around the house and the area the Evotouch is in - you might be surprised to find it isn't working very well due to interference.
I do have a couple of cameras that are 2.4GHz only, but they normally never get disconnected. It's true that the Nest was dual band, though.
By the way, how do you know which firmware is installed? I can't see it anywhere...