If you don't believe .0...1 exists (infinite 0s followed by a 1) then this ought to convince you that .9... == 1 (as (x+y)/2 == x => x == y) but since we're (hypothetically) arguing with someone who believes .9... != 1 it's quite plausible that the person is also going to believe in .0...1 existing, also.
What I suspect is actually going on (I've also met otherwise-savvy people who will debate this issue also) is that the human numerical-cognition system uses something like a crude form of nonstandard analysis internally; and, under this theory, you see this internal system (and its intuitions) leak out in some people smart enough to get that .9... really does have 'infinitely many' 9s but who also for whatever reason are both undereducated on this point of mathematics and generally self-assured.
Briefly: you can extend the real #s with a new entity (call it delta) such that:
- delta > 0
- if r is a nonzero # then delta < |r|
- (for now) delta^2 = 0
(in effect: delta is an infinitesimal, it's nonzero but smaller than any other positive #, and delta^2 is 0 is mainly an algebraic convenience).
Once you've extended things by including delta you can write extended #s in the form r + s x delta (eg: 1 - delta).
If we re-define ".9..." to be 1 - delta, then the following all drop out of that:
- .9... != 1 (1 - .9... == delta != 0)
- .9... < 1, (1 - .9... == delta > 0)
- and we even get: (1 + .9...)/2 = 1 - (delta/2) ==> (1 + .9...)/2 > .9...
And more generally in my experience most of the spinning-in-circles around people with beliefs about .9... != 1 is that the people who vociferously believe .9... != 1 don't even know about non-standard analysis (or infinitesimals, etc.) and thus can't articulate what their intuition is telling them, but what their intuition is using is basically (a nonrigorous form of) analysis with infinitesimals.