Technically, altering the document creates a counter offer- the original offer was rejected. If you make a counter offer for employment, and they behave like that offer is accepted, (ie by proceeding with onboarding, sending paychecks and assigning work,) it was accepted. Maybe not everywhere.
I’ve a friend that does this all the time- specifically both arbitration and non compete clauses. Just because they use standardized forms doesn’t mean the contract isn’t unique.