Just on the subject of incarnations of the Doctor being linked by regeneration, I feel it should be noted that we currently don't know if the pre-Hartnell Doctors became the First Doctor through regeneration or not. The Timeless Children states that the Doctor's childhood was real, so therefor not a memory implant or lie or anything like that. So it's possible that the Doctor regenerated into a child at some point and had their mind wiped (or the other way around, it doesn't really matter). But it is equally possible that they were turned into the First Doctor by some other means.
I'm just saying it might be a good idea to hold of on using regeneration itself as a measure of placement, at least until more of the Timeless Child storyline is revealed. There's a gap in the narrative at the moment, and it could very well turn out that there is a non-regeneration transition separating pre and post Hartnell incarnations.