It depends on where you want to start your axioms. We can go the Whitehead/Russell route or just use this as an axiom.

We convince children that 1+1 =2 without delving into the Peano axioms. It's ok to not delve too deeply into the axiomatic structure of the reals.

You're just proving their point. None of the things you talk about are even remotely obvious or “natural” to people we're talking about. You want them to ”choose axioms”? Axiom-a-whaaa? 1+1=2 does not need Peano axioms because it's cognitively fundamentally different than 0.999...=1.0. It is immediately obvious because dealing with simple arithmetic on natural numbers is in everybody's experience constantly. In other words we do not need to convince them. Clever untrained people would be able to get 0.999... thing quickly if you gave them some explanation and let them think a bit, if they cared, but properties of real numbers are far from obvious or basic.

> Okay, so it follows from completeness.

There is nothing to do with completeness here; 0.999… = 1 is a statement about a series of rational numbers summing to a rational number, and the convergence of the series is established by the fact that it sums to the right-hand side, not by an abstract appeal to completeness.