Anyhow if I'm correct, this form of reasoning for it to stand the
quest, then there must be a clear line of justification for it. For
instance the argument about whether the reals are countable actually
means literally whether there is a bijection between the reals and N,
so N is at the heart of the subject. Now to go and say that
countability of the reals (which means bijectivity of reals to N) can
only be reached about by circumventing N and using another countable
infinite set N* as the domain for any sequence in an argument using
limits is really strange somehow.

What you are having is the following:

[1]When we use N as the domain of injections (x_n), (a_n) and (b_n),
then Cantors argument PROVES and SHOWS that there is a real that is
not in the range of those functions.

[2]When we use N* as the domain of injections (x_n), (a_n) and (b_n),
then Cantor's argument will seize from working in the same way to show
the missing real.

[3]However we also have the corollary that even when we use N* as the
domain of those functions, still we can by a single common well
defined way define another sequence with exactly the same range of
those functions but from domain N, and we can apply Cantor's argument
and SHOW a missing real in the rang of those functions!

Now you call [1] deficient, [2] apt to reality standards [3] bogus.

Why? because we used N in an argument that involves a higher order
concept that must use N* instead. (That's your reply).

But again: why? what is the higher order part of the argument that you
see it demanding circumventing the heart of the subject (which is N
really) to some N*.

Is it the definition of Limit.

But limit is defined in this argument as the least upper bound, and I
don't see in the definition of L that I wrote (which is the standard
by the way) anything that has to do with necessarily picking it up
from some Omega_th end point? that has no meaning at all, so why?

Should I adopt this rational of yours then I'd ask you: why not say
pick L from the -1_th starting point. i.e. choose your domain to be
{-1,0,1,2,3,...} since this clearly also preclude Cantor's argument
and you clearly can make L be the -1_th digit of (x_n) [Remember a_0
is x_0, so x_{-1} lies "before" a_0].

Or you'll say that {-1,0,2,3,...} is also a kind of high order
countable set?

Your argument is simply shunning one of the most important two sets in
this argument, that is N, and using some replacement, without any
clear justification.