the set of normal numbers in $[0,1]$ is measure 1. Which means the non-normal numbers are a null set. Of course, contradictorily, the set of numbers which we can prove to be normal is probably countable.

So as I suspected my claim was (technically) wrong, but because most numbers are normal, not (as I thought) because most numbers aren't.

Your claim is correct as written, I think.

It is the case that for most numbers, it is unknown if the number is normal. (That is, for a real number between 0 and 1 chosen uniformly, the probability that we know whether it is normal or not is 0.)

It is also the case that we know that most numbers are normal. (That is, for a real number between 0 and 1 chosen uniformly, the probability that we have selected a normal number is 1.)

English is not great at highlighting the difference between the first claim and the negation of the second!