Considering that people were having typical days of programming before complexity theory existed, of course it's not required. I don't know of anybody who seriously claims that it is required.
This is my field of research, and I assure you computability theory is not a subset of it.
By that metric (and even by your own argument "the inability for algorithms"), the undecidability of the halting problem definitely fits within the complexity definition.
> imposing restrictions on the available resources is what distinguishes computational complexity from computability theory
And for the record when I say the ability of algorithms to do something I mean the class of all possible algorithms.
In any case, it's clear from your appeal-to-wikipedia that this conversation is over.
Because you are not constraining any resources.