1. He'd Inspire Curiosity
Just consider how he held his older self's attention at the end of 'Day Of The Doctor'. Wouldn't you feel lucky to work with him down in the UNIT galleries? He seems a man full of wonderful mystery. Which could easily inspire whole generations to take more interest in art and indeed galactic affairs with a well-timed question, phrase or simple words of encouragement. Perhaps he'd even snaffle jobs in his department for past/future selves. We can just imagine the First Doctor as the grumpy caretaker, the Second providing recorder accompaniment to audiovisual installations, the Third, Sixth, Ninth & Twelfth on security detail, the Fifth, Seventh & Eighth as handy fellow guiding colleagues and the Tenth & Eleventh as young, strong painting movers. You'd have to conclude that he'd be a brilliant boss to work under. Relaxed but still able to get the important jobs done on time and efficient enough to marshal colleagues into achieving even by unorthodox methods ( witness his various speeches to companions in the years from 'Robot' to 'Logopolis'), he's got everything you need to be a great curator. Have another jelly baby, and feel free to send in your suggestions for pieces that should take pride of place in the Fourth Doctor's gallery!