Which is probably the key intuition here - the expected number of matches is exactly the same, but because you can have more matches for the purely-repeated sequence in the same string the total number of strings containing those matches is smaller.

Sure, but the question is "how likely" not "how many times". Counting the expected number of times is a much easier problem because you can use the linearity of expectation.

But you're trying to find a bit-substring of length 2 that is least likely to have been transmitted.

If the bit strings of length 3 are chosen uniformly, 11 is less likely to appear than 10.

You are totally right. He is incorrect.