Writing "this is a contract of adhesion" basically means "we gave this contract to the applicant and asked them to take it or leave it".

A court will be much more favorable to the applicant, then, because presenting them with a contract they have little power over is something courts generally don't like, as it is unjust.

Asking the employer to write "this is a contract of adhesion" basically means "oh, you're screwing me over? Can you please write that this is a screwing-over contract on it?".

Contracts of adhesion still are still pretty solid. Generally, it's only going to get voided if the process (i.e. take it or leave it) and the terms are unconscionable.

Courts uphold shrink wrap and click-to-agree contracts all the time.

I think it would mean that it would be more likely for a court to scrutinize the terms of the contract with an eye toward whether they were reasonable and fair, rather than simply enforcing it as written.

(ETA:) This might not help very much, because in places where noncompetes are standard, a court would be sticking its neck out a little to rule that one is unreasonable. But maybe it would help a little, if one couldn't get the provision struck from the contract altogether, and had no reasonable alternative to accepting the job. (Again: IANAL, and I'm just speculating -- no one should take this as legal advice!)