I freeze pizza dough, but I wouldn't recommend it. Maybe letting it rest after it thaws is a good idea, re gluten/elasticity. (I'm usually in a hurry),
The thing is, when you read recipes or watch videos, making pasta dough seems like a big deal. But I learned from my italian teacher (from Padua and Bologna, and those folks know what they're doing) and she just threw an egg and some flour together, did not even knead other than to make sure it cohered, and the pasta came out swell by the time it passed through the machine to enough thinness. I think that if you want to make batches of, say, fettucine, dry them on a rack or on hangers, and when dry, put in packages and store in a cool and dry place... and that doing that with a bigger batch of dough is a better idea than storing the dough in the refrigerator or freezer.
(opinions may vary)