To be a bit of a jerk he was always an alien.
I get what you mean, by more of a divine not able to die situation, but I am not sure if that was defined totally yet. Just that there were infinite regenerations.
And we know that Time Lords can die, even if on regeneration 1. The rules do not seem super well flushed out but if you blow them up for instance the pieces are not going to regenerate. Seems like life threatening injuries are of varying degrees useful though.
Daveym has gone more into on here than I have. Regeneration has for sure change and it is more of a superpower now. Even before the revalation. The reveal of the timeless child just made it more pointless to care about the character that much and more alien.
Really no need for the Dr to ever have to regenerate, as seems to functionally immortal!