Turing completeness is the upper bound of computability, not the lower bound. It's useful mostly for showing that some thing can express the full range of computable problems, or for snarking that some thing is far more complex than it has any right to be.
Total languages omit partiality and non-termination from Turing completeness.
Partiality is IMO irrelevant when it comes to computability. Any partial function (that is, one whose range is not defined over its whole domain) can be expressed as a total function by either constricting the domain or expanding the range. For example, a "pop" operation on a stack is not defined for an empty stack. You can just loop forever if pop() is called on an empty stack. Alternatively, you can require that pop() is given a witness that the stack is non-empty, or you can require that pop() returns either the top-most element of the stack or a value that indicates the stack was empty. Both let you compute the same set of things as the former.
Non-termination is required to be Turing complete, because being Turing complete means being able to compute functions that one cannot reasonably expect to complete before the heat death of the universe. In _practice_ every function terminates when the computing process dies due to some external factor: process runs out of memory ("real" Turing machines have infinite memory!), user runs out of patience, machine runs out of power, universe runs out of stars, that sort of thing, so _in practice_ doing 2^64 iterations before giving up will generally* give you the same outcome as doing an unbounded number of iterations: it'll either terminate, or the process will be killed (here, due to reaching its iteration limit).
On the flip side, giving up non-termination and partiality only gives you increased correctness. If there's one thing we've definitely established in computing, it's that we will readily discard correctness to gain a little extra productivity. Why make a developer implement code to handle reaching an iteration limit when you can just make the user get sick of waiting and kill your app?
* 18 quintillion is a very large number. Have a try. The most trivial recursive function, on my M4 Mac, when convincing clang to be smart enough to turn it into a loop but dumb enough not to elide it altogether, would take a bit shy of 600 years to complete if iterating ULONG_MAX times; I didn't wait for that, if I'm honest with you, I ran it with a much smaller iteration count and multiplied it out.