You're having very common feelings. There
are people that push their children and want to believe that they are gifted when they are likely just advanced. Most people that find this board however, find it because they have the mindset of "I suspect something may be up with DS/DD and I need to figure what to do." Your skepticism is healthy and shows that you're trying to look at the situation objectively.
Just so you know, I went from "DS's average" to "DS's probably gifted" to "seriously, DS's just average" to "DS must be PG". Just search the archives for my posts

In our case, getting DS tested didn't tell us the complete picture (in some respects we have more questions), but it did put our mind at ease. We got independent validation of what we suspected.
I'd recommend getting him tested. If you have the means to get him tested yourself privately that gives you the option to decide if you want to share those results with the school. (You might want to inquire about which tests that they give because you can't re-test using the same test for a period of time.)
Good luck.
JB