Toilet tank will not refill after flush unless water valve is closed and reopened

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..

Septic System?? If so, the system is FULL of water already and needs to drain out from your drain field. If it is a municipal system, tree roots CAN SWELL TO 40 % or more of their normal size in wet weather, creating a blockage- hence a slower drainage- I'd look into a rooter service if this is the case and if on a Septic System- even pumping the tank will not help as the ground water is the culprit-preventing more water to disperse. Pumping the tank will only remove liquid & solids in the inlet side- not removing water from the leach/drain field.