You have hit the nail on the head. Your proof is correct. Here are some clarifications:
Let be a finite list of primes of the form . Let . Since , cannot be prime, so it is composite. When divided by any , it leaves a remainder , therefore is not divisible by any prime of the form . So must be the product of primes only of the form . But if you take the product of numbers of the form , the product is also of the form . So must leave a remainder when divided by . But by construction, leaves a remainder when divided by . We have reached a contradiction, therefore our premise is false: there must be an infinite number of primes of the form .
This argument does not work for the case. Can you figure out why?