if you put 1 out then there is no reason to believe it is uncountable because you can sort the list such that the diagonal is all 0s (if you know an argument that shows [0,1) is uncountable I'd be happy to know)
It is a superset of (0,1), which is uncountable because it is an open interval. All open intervals except the empty one are uncountable. What kind of proof would you like?
There is also no such thing as an uncountable list. Lists can be indexed by integers, which makes them all countable.
the proofs that I've read shows that the list of infinite binary strings between [0,1] are uncountable because the invert of the diagonal of the list is not on the list, do you know any proof that does not refer to the invert of the diagonal?
you can't prove something just by assuming it that would be trivial, if we can build or convince ourselves how to build a wormhole or a time machine then we can believe in Real numbers
What Koshkin says is correct and has already been proven by others. You want to get a book that covers sets, maps and cardinal numbers, and read it carefully if you are interested in this sort of stuff.
I can see that you can symbolically talk about the power set of (0,1) and say it's cardinality is bigger than the set itself (although I haven't studied sets and cardinalities deeply enough) but I can't see what Real numbers offers against the computable Universe hypothesis
There is also no such thing as an uncountable list. Lists can be indexed by integers, which makes them all countable.