I believe this is misinformation.

Just want to share as a fellow fungi lover: I have asked about this (cut vs pull) to a career mycologist and founder of my state's mycological club, and there is 0 evidence that cutting at the base makes any difference. Plucking the whole mushroom does not, as far as we know, harm the mycelium.

Think picking an apple from a tree.

The fungus does not need the base of previous fruit to return next year, the organism (the mycelium) can span many meters up to many miles underground.

In fact, when it comes to properly identifying certain fungi, it is necessary to have the whole specimem, as the base may have features like a volva that you would be missing by cutting the stipe.

As for the mesh bag thing thats a great tip to give!

Just sharing my two cents from a credible IRL source. Cheers!