I've bought a few of these before (no affiliation) amazon.com/dp/B0CCL7TJ48

Edit: this is a link to 1.5V rechargeable batteries, which I commented before OP's edit acknowledging them.

It boils down to chemistry. The nickle-metal-hydride chemistry used in rechargeable batteries just don't produce 1.5V (explaining why is a lot more complicated), while the alkaline chemistry in non-rechargable batteries produces slightly more volts.In the same vein, lithium-ion batteries produce around 3.6V, while lead-acid produce about 12.4V.

To answer your other question, they should be able to power most things that aren't an analog clock (which might run slightly slow).

classic rechargeable batteries are always 1.2V. That's due to the chemistry.

The "low power" warning is the reason why a lot of devices will tell you not to use rechargeable batteries, as they will trigger the low battery warning earlier, and the cutoff voltage might be reached earlier.

I think there are modern rechargable lithium batteries with usb ports that will emulate 1.5V batteries and their voltage curve. But they have their own quirks as well

Different chemistries produce different voltages per cell. Alkaline produces 1.5v, NiMH produces 1.2v, Li-Ion produces 3.6v. These are averages, the actual voltage varies over the current charge level of the cell. This variation in voltage is how the low battery alarm actually works, although Alkaline cells produce 1.5v initially, once they are nearly empty they are producing 1.1-1.2v. Your thermostats will likely work fine on NiMH batteries, if you can live with them continually complaining about the batteries being low.

There are, or at least were, rechargeable alkaline batteries, but they don't last many cycles.

The 1.5v Li-Ions have a tiny circuit board on them that regulates the voltage down to 1.5v, which takes up space so the capacity is reduced. You could do that with NiMH, but it would have less capacity than the Li-Ion version, so there's little point.

Eneloops are about the best rechargable AA and AAAs you can buy, they last close to forever. I've got some that are a decade old and still in use after thousands of uses and still about 80% of original. You chose well.

And it's fine. I used to use them a lot for GPSrs. A pair of eneloops lasts about 8 hours in an Oregon. A pair of decent branded Alkaline AAs lasts about 7-8.

The voltage is not an indicator of charge life when you're comparing different chemistries. The reason you're seeing low charge is that your device is not calibrated for rechargables. In things that are, they have settings for both so you can select which you use. If your thermostats don't, then they'll always removed about anything other than Alkalines.