That's no proof. What I think is that they're born with the mushrooms (I guess they can have different designs) and could take them off, but they're Toads and it's a big part of them. Normally I wouldn't think this, but considering that they never say it can't come off, but in some sources (even if only approved by Nintendo) it can, so I guess it'd be smart just to stick with that. I mean, why would they all wear similar hats (and some females have hair, so don't say baldness)? Oh, and it could be like their turban for all we know...
Do the Mario! Swing your arms from side to side, come on, it''s time to go do the Mario! Take one step, and then again, let''s do the Mario all together now! You got it! It''s the Mario! Do the Mario! Swing your arms from side to side, come on, it''s time to go do the Mario! Take one step, and then again! Let''s do the Mario altogether now! Come on now! Just like that!