I mean, the first priority of Time Lord fashion is to create personal space. Is the robe and the collar/hat voluminous enough to act as a force field, and compel everyone to stand 4-10 feet away from you at all times, to discourage any inadvertent to*ching or advertent h*nd h*lding? Welp, better make it bigger, just in case!
The second priority is maximum plumage and aesthetic ridiculousness, because while they don’t want anyone to to*ch, they DEFINITELY want everyone to LOOK. Big dumb collars, big dumb hats, shiny dumb skullcaps, and glittery robes – they bedazzle the shit out of EVERYTHING. I mean, even the CIA robes – which are supposedly some of the simplest, least fussy robes on the whole planet – are made of sparkly, glittery fabric that puts Twilight vampires to shame. Look at this disco situation:
I think they’re wearing shirts and trousers under there because then they can unfasten the front of their robes and let them billow out behind them like a long coat and look really cool while still technically being decent and not naked