There are only 2 smart thermostats: ecobee and nest. Looks like you bought an older house so the wiring could be an issue.
Why would you worry about the wifi signal going out? You buy a nest or ecobee to run themselves, you don’t touch them unless you want the windows open. Who cares if you can’t see the thermostat for hours or days. Keep your hands off of the controls. I’ve owned ecobee thermostats in my last 3 houses and I prefer ecobee over the others.

My system, and many others in The Villages, do not have a reheat function. So, they are not true dehumidifiers. They reduce humidity by "overcooling" the house to reduce humidity. So, if you set the temperature at 78 degrees, and the humidity at 60 percent, the system will cool the house down to about 75 degrees when the humidity rises above 60 percent. This is a compromise that works fine for snowbirds who want to save a few dollars on electricity by controlling the humidity with a higher than desired temperature. But, if you are living in the house, you want to maintain a comfortable temperature that stays at the setpoint. That is why I turn off the dehumidify feature on the thermostat. It is an optional feature. If you want to control both the temperature and the humidity, you need a true dehumidifier with a reheat function.

Is it possible you have 2 units, the main furnace/ac combo and maybe an AC unit for an enclosed lanai that just does demunidifying??? People put in these split units and permit them as dehumidification only as adding AC is building an addition to the house… or something like that.

Just had an ecobee installed. I gought it and good guys installed for $150. I would have done it myself but you are correct. Carrier had just 2 wires. HVAC guy had to go to ait handler and allatch other wires. I just didn't have the skills. Love the ecobee and yes, if you lose power, wifi will come back on as will ecobee

Have used Sun Kool in both homes we have owned and they have done an excellent job for us - I would suggest a Lennox system as we have had no problems when the wireless goes down and then comes back up - the system re connects and away we go - back to normal!!

Thermostat question
 

I had installed a Honeywell thermostat model TH9302 0WF 5003 when we bought our property. it is a Wi-Fi thermostat so you can look at your settings and also temperature and humidity when you’re away using your app. When I’m not there, I typically keep my house between 80 and 82° and my humidity is around 35 to 40% inside the house. if your Wi-Fi should go off, I am pretty sure the thermostat will stay at the current settings and maintain temperature. I have had my Wi-Fi off a few times and everything came back on the way it should. I’m not sure what the Honeywell humidistat is on your wall, I would not think you’d have a Humidifier that you would have used in the winter time but not sure. Typically, that is what that humidistat would be used for. These thermostats are typically not hard to install and also not hard to set up but that depends on your comfort level. Good luck with your search.

My husband is an electrician. We use the humidistat while we are in NE for the summer. We sent it at 85, the air will only come on if the humidity is high, not the temperature. We like Munn’s the best.

If you are a seasonal resident, one of the best things you can have is a reliable handyman or neighbor who has access to your house when you are not there. If there were an extended power outage, or any other potentially harmful event, there are many other things besides rebooting your wifi that may need eyes-on attention.

Inside climate control
 

I have the same set up that you have and are also away for six months every year and it works perfectly, A/C hardly if ever comes on. I also have a separate humidistat which is battery powered and a WiFi camera focussed on that for monitoring. If the WiFi goes out the cameras are the only ones affected.

Running the compressor more by lowering your temperature setpoint will remove more water vapor and send it down the drain but may or may not lower relative humidity. (both the numerator and denominator are lowered).

Does it lower your relative humidity as measured by the thermostat or HVAC controller?

I have programmed commercial controls for HVAC. Heater controls are based on temperature and relative humidity. No dehumidifiers needed for human comfort in most regions. They are needed for controlling the environment in sensitive areas such as a laboratory or computer/server rooms.

I assume that it lowers the relative humidity or the AC would never shut off. As I said, I never use the dehumidify feature because the relative humidity in my house is always below 60 percent, and usually below 50 percent. The only advantage I can see for using the dehumidify feature is to raise the thermostat setting to 80 degrees or higher while the house is unoccupied to save money on electricity. But ideally, to control both the temperature and the humidity, you need a reheat function, which many HVAC units do not have.

I have an Ecobee 3 Lite. It is wifi, Alexa, and Google enabled. It also has a humidity control built in, letting you set a max humidity allowed before the AC turns on to remove it. It also have the rest of what you expect in a thermostat (vacation mode and such).

It keeps its setting after a power loss. For my Internet router, I have a UPS to keep it running for 2 hours worth of power loss.

https://a.co/d/9ckyc1R

I agree. Most homes are well insulated and do not need reheat.

interesting, but doesn't explain why my unit rarely shuts off even though it shows the temp to match what it's set for for hours on end.

What you have is very old technology. We had one in our house from 2014. Now with the new Smart thermostats it is a much better and easier process. For years we had a Honeywell Smart thermostat and liked it. When we replaced our AC unit this year they installed the Ecobee thermostat. I like it better than the Honeywell for many reasons and it has a lot more features and is very easy to use. So yes update it. Also we have been using Kalos AC company for many years and the always provide excellent service. When we replaced our AC unit we got quotes from all the major venders in the area. Mounds and SunKool were a lot higher. Kalos and Sunshine AC were very close in price.