My toilet won’t refill. On flushing, no water flows through the refill tube into the overflow pipe (though a little bit drips onto the float cup). The external water valve is open, and there’s no sign of leaks around the valve or the water line. If I refill the tank with a bucket, the tank remains full and the flapper is secure. But if I flush, same problem – no water from the refill tube. Should I replace the whole mechanism inside the tank? Particular parts? Or is my problem with the external water line and valve?

You have a ball cock with a cup and not a float ball. Is that correct? That would be a fluid master ballcock that sounds like you got some trash blocking the flow.
Turn the water off to the toilet and open the lid. Twist the top of the ball cock counterclockwise and pull up. Remove the black washer and place a pan over the ball cock. Now turn the water on for a few minutes flushed everything out. Reassemble and test. Is it better now? Good luck, Tom

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Twist the top of the ball cock counterclockwise and pull up.

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You have to lift the little lever up to do that.

Close the angle valve remove the tank lid remove the cap of the fill valve as Tom says put your hand on top of the fill valve, now open the angle valve see if the water is flowing if it does do as Tom says if not problem is with your angle valve, Good luck.

John