Some people after withdrawing tend to tighten up their play subconsciously with a smaller roll, this will have an impact on results.
Thats actually a really good point and might be part of the explanation for this myth, which has excisted since the birth of online poker. Few people move down after withdrawing money, so withdrawing always mean, we become more aggressive with our
bankroll management. This can then lead to us being overly focused on min-cashing in MTTs, because now we kind of need that money to maintain our bankroll. Or if we play cash games, maybe we shy away from high variance but profitable spots and play with "scared money".
For a lot of people the path to tilt is also shorter, if even a modest downswing might force us to do unpleasant things like redepositing or moving down. Its much easier to play well, when we are overrolled for the games, we play. So ideally we should not withdraw funds, if we are serious about online poker and eventually want to make it into mid or high stakes games. That being said I have withdrawn most of my winnings myself, because.... well... life happens.