The amount of focus on the subject itself which is required to do mathematical research of the highest caliber is very high. It's plausible that the focuses entailed by vanity and ambition are detrimental to subject matter focus.

This rings true in my experience as well. I've found that thinking about how other people will perceive what you're doing tends to derail mathematical thought. It would be interesting to look at other fields where excellence requires extreme focus and see if they also exhibit the same phenomenon.

aspiring mathematicians would do well to work to curb their ambition and vanity and increase their attraction to mathematics for its own sake.

I think this is good advice. Combining it with Grothendieck's idea about the importance of complete innocence gives you

Aspiring mathematicians would do well to work to curb their ambition and vanity, and increase their playful innocent attraction to mathematics for its own sake.