Sounds like an indirect TPMS. These can be easily damaged by tire-mounting machines. If one or both are damaged, you can get dash light staying on. If they're not damaged, sometimes they just need a reset, especially if you don't change all tires at once. Indirect sytems test revolutions, not pressure. So it will sense a problem if 2 front tires have more wear than 2 new back tires. Reset procedures vary, sometimes you need a scan tool. You have be sure all 4 tires have the correct pressure before resetting for wear. If you have a driver display interface, sometimes they can be reset from the menu. Like Fred Wilson said, the batteries do wear out after 10 years, +/-. But, like you, I also think that's way too coincidental. I'd get a second opinion to make sure they're not damaged, check all pressures and see if they can be reset. Indirect TPMS will be on valve stems or attached to a wheel's drop center. Good luck!
Edit for your questions in the comment section. It wouldn't let me put this as just a comment so I had to edit my answer instead. Let me ask a couple of questions. Can you confirm if it's an indirect sytem for me? When you say it flashes sometimes, how so? Does it flash like 3 times and go solid again? How often? Does the ABS light come on? When they showed you the device, how do you actually know the fault was low batteries? What did it say or look like specifically? Also, some models have a reset button under steering column where your knees are. Look to see- do you have a reset button? If so, don't touch it yet, just let me know. Did you check the tire pressure on all 4 wheels yourself to make sure they're correct? It's possible you can use an OBD2 code reader to erase faults but, if you could, I'd like to know what the fault codes are if you do use it. It should connect to the instrument panel. A light on your dash can be from some type of damage and will give a code. Sometimes a relearn process needs to be done by the service technician after servicing your tires. If it's not done or done incorrectly that could also cause a fault.
Edit for additional comment questions: Is your reader wireless as well? More details on how you did your reset- something like, held it in for 3 or 4 seconds and the tpms light blinked? Is that about right? How long did it blink for, how many times, how quickly? The lowbat indicator you spoke of: that was from the serviceman's wireless tpms tool, right? He came to the front to show you this AFTER the work was done? Odd that he wouldn't scan each sensor FIRST to get info on them, including preexisting faults and to get their custom ID's so they know how the ECU recognizes them. Can you give me a little more detail about when your tpms light blinks? Only on startup? While driving anytime at all? If so, what details can you give me about that? If your OBD2 reader is wireless, you should be able to verify codes for low batteries or anything else yourself. Have you? Lastly, the ECU stores where each sensor is for each tire. It may be worth a try swapping the back two tires, filling them to spec and redoing the reset (just in case they swapped those two wheels by accident when they were doing the work). On a side note, those trouble codes you did mention- P0420 and P0116 together indicate you should check your coolant temp sonsor with a multimeter to see if it's bad or check your thermostat.
Updated edit: Most direct systems have some way of knowing what tire's sensor it's reading from. Sometimes it's by unique SN's or ID's, sometimes it's specially placed antennae that detect directional rotation; but, the point is, the two back sensors could have been mixed up. Which would be fine if a resynchronization is done (recalibrating the transponders). What you did with the reset button was reset the tire pressure baseline. I heard at some point that some models need a wireless tool to resync with Toyota models. If you did the pressure reset correctly- Hold the button with the ignition on (not car on) until the tpms light blink 3 times, then leave ignition on for a few minutes before shutting it off- then it sounds like you did it right. Since that didn't work, I would try swapping the two back tires to see if ID's were out of place. Lastly, bring your car into a different tire service place to explain the situation. Ask if they can try resynchronization first, since you suspect it may not have been done. Then ask if they can get any trouble codes to verify low battery on BOTH sensors. Maybe they'd be willing to look and make sure nothing was damaged like an antenna or seal... Finally, if none of that works out and their codes match low batteries, 'I guess' that's it but would be very coincidental for even just one to suddenly have become just low enough at the time of servicing one tire, let alone both!
Regarding the other codes- the ECS can be tested with a multimeter. An auto part store should have spec values for that or your manual. If your thermostat is stuck open, you'll feel coolant running through the upper radiator hose upon initial startup (it shouldn't open until it warms up a bit so you shouldn't feel fluid running yet). It'll also take longer than normal for your heat to start working and your engine to warm up. If it's stuck closed, your engine will run hotter than normal and overheat. Those two codes you got could also just be from letting your coolant level get too low at some point. Clear them out and keep an eye on your engine temp. If you get any warning signs then you can see if you get those codes again. If so, one of them is probably defective. Sorry I couldn't be more help. Good luck.