By that point, the paperwork is complete. The hiring completes the transaction as being satisfactory to both sides.

That's not generally how it works. Modifying a contract without being clear that you are making a counter offer isn't going to hold up in court. They are going to see the company's offer, your acceptance, and then hold you to the original company offer.

Fwiw, I'd say at least half of the clauses I've been asked to sign were not included at contract time, but rather at post-contract "do HR paperwork" time.

So I don't see how that could be construed to be 'part of the original offer'.

Especially if you asked (in writing) if such a clause existed and received a negative / no response before signing the contract.

Depends. There could be a reference in the main contract listing execution of the non-compete as a condition of the validity of the contract.

I'd still prefer to have issues resolved before I start employment somewhere, rather than having to deal with it when you're already at work and walking away is a bigger step.

For the vast majority of workers the company has the right at any time to introduce a new contract and require you to sign it as a condition of continued employment.