Let's get naming conventions right: the "Doctors" thing as currently in use is not, strictly speaking, an infobox. It's more of a navbox, although more properly, it is what you are proposing which best fits the definition of "navbox". (The "Doctors" template, as we know it, is a weird, one-of-a-kind hybrid of a navigation template.)
At any rate, there was more than one proposal of a navbox with all the Doctors before, and I believe one was shot down because of concerns that it would render Special:WhatLinksHere much harder to use for the relevant pages — e.g. trying to find every page which has something to do with The Doctor (Party Animals) would rather spuriously direct people to pages like The Relic (Alien Bodies). Besides, a navbox such as you are proposing can only be put at the end of an article. It might possibly supplement the "Doctors" template located below the infobox, but it serves quite a different purpose; the small, discreet Doctors navigation box below the infobox is a much handier tool for quick navigation by casual readers.
Anyway, @User:Epsilon the Eternal, we can't exactly hold off on doing this forever because two or five seasons from now we might hear more about Tecteun. The sources which do exist give us a dozen or so new Doctors to reckon with, and we have to deal with it now, even if we have to change things again later.
For myself, I think the WhatLinksHere concerns are perhaps a bit overstated, and wouldn't say no to a complete navbox, but only in addition to a revamped version of Template:Doctors similar to what has been proposed above.