Just jumping in here, there's a few things that I want to make my voice heard on. I've read the thread and correct me if I'm wrong, but there seems to be two types of groups here: Those who want to focus on the new and those who want to lean more to the established. Both approaches are perfectly valid, but make sure to leave bias at the door. Remember, we have 50+ spinoffs and like hundreds of Doctor Who stories, the continuity is always going to be shifting and overlapping.
Secondly, I do like the second infobox template that Memes showed. But, I'd also like to propose a bit of a radical change: Dump the incarnations infobox entirely, and use this instead.
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This is just a placebo, of course, and there's a lot more Doctors we'd need on this thing. But in this format, we could include all of them and not have to have an argument over an infobox.