Why did the fill valve in the toilet pop up

Run enough water to get the hot water heater to start its cycle, then close everything and measure the pressure until the heater gets to the end of its cycle. If you have a check valve and no expansion tank, this can cause a pressure issue that wouldn't be detected with a quick test. Instead, it will slowly rise as the water heater runs, and then drop back to normal when the first faucet is run.

It's also possible that all the toilets have the same parts and are therefore failing at the same time. But before replacing everything, it's worth it to make sure they are really faulty.

It seems as if you have an obstruction that may be floating around in the pipe or valve. Sometimes it obstructs sometimes it flows clear. I would start with changing the water valve. If this is the old style "gate" valve there are several things that can clog the line. I would replace it with a 1/4 turn ball valve. That will give you much better flow than a gate valve. You will need to shut down the water probably at the main. The old valve is probably soldered, but you could cut it off and use a compression type valve instead. "Sharkbite" is one such supplier. Although I have never used their compression valve, I have used their end caps and found them to work well.

If you can have someone hold a bucket under the pipe once you cut off the valve and then a second person turns on the water main to fill the bucket to purge the line of any debris. With cast iron pipes it is common to get clogs from rust build up. The rust may not pass the small opening in the old gate valve when you first purged the line.. Although you did not mention what type of pipes you have, if it's copper than you won't normally find rust obstructions..