The origin of the Doctor as the Timeless Child may be a clue as to why they chose that moniker. Tecteun constantly experiments on them, and how would Tecteun make this acceptable to the Child? By saying the Child is ill and "needs a doctor". Their whole childhood revolves around the idea of a doctor making someone better.
The Timeless Child was a young girl picked up by a stranger and travelled the stars with them - much like how the Doctor would take companions with them to do the same in their later lives. The Doctor was the first Companion.
Supposing the Curator actually is the Doctor as is so heavily implied, this would mean he's fully aware of the Doctor's true nature and his comment about revisiting the old favourite faces would certainly imply he knew the Doctor had more regenerations to spare just going back and choosing old faces for the fun of it, even though as far as the Eleventh Doctor knew at the time he was on his last life.
Far from lowering the stakes of the show by giving the Doctor infinite regenerations, the episode actually raises the stakes. Since we're never told that the Timeless Child has infinite regenerations, they could run out at any point. Even the second set of twelve from "The Time of the Doctor" isn't guaranteed — the Time Lords could simply have lifted the limit they placed on the Doctor, rather than giving him extra lives. So, from now on, any of the Doctor's regenerations could be their last, and they have no idea which one!
Fridge Funny: The Doctor escapes from the Matrix by overloading its systems with a super-accelerated flood of her own accumulated memories. In effect, she forced it to binge-watch the entirety of Doctor Who in a couple of seconds!
The Timeless Child undergoes many regenerations during Tecteun's investigations. We can only assume that the deaths are caused by Tecteun's experimentations.
The only question is: were they a consequence of said experiments, or did she intentionally cause them as part of said experiments?
If the point of the experiments was to figure out how the Child regenerated, then it's almost certainly the latter. That's an incredibly traumatic, cruel thing for anyone — let alone a parental figure — to inflict on a child.
The fact that the Doctor has had entire lifetimes erased from their memory makes one wonder how many companions the Doctor had that they now have no knowledge of.
And why the Doctor periodically decides not to get new ones.
In Series 13, the Doctor gets to meet one of their past companions that they don't remember. Said companion is understandably hurt by the fact the Doctor knows nothing of their time together.
All the way back in "The War Games", the Second Doctor indicated that Time Lords had an unlimited number of regenerations unless their body were damaged too severely, in what seemed a throw-away bit of Early Instalment Weirdness. Fast-forward to this story, and one can't help but wonder if the memory block they had placed on the Doctor was gradually starting to wear off — and whether or not their exiling him to Earth was motivated in part by ensuring he'd be stuck somewhere he wouldn't be able to get revenge on the Time Lords in the event that the memory block failed completely.
The Fifth Doctor started to fade away when his previous selves were taken out of time. He said, "A man is the sum of his memories, a Time Lord doubly so." Imagine how many lifetimes of memories have been erased or suppressed, compared to just four in that episode!
