I do not carry my child's birth certificate with me and unless Disney said I should bring a copy along to prove her age I'd be pretty upset if that happened to me too. In fact, we don't even have her birth certificate cause it was stolen (along with mine and my husband's, our marriage certificate, ugh, I could go on) when my mom's house was burglarized last summer. We have not gotten her (or ourselves) new ones as we have no need for them at this time. We each have passports but never bring them with us for domestic travel so we have no ID for our daughter at all when we're on vacation. If they told me when I book a trip that I'll need ID for my 2yo then I'd bring her passport, but if I'm not told that, why would I carry ID for a 2yo? So in response to the OP, if I wanted to sneak in my daughter once she turns 3 I'd just use that same reasoning, that I was not told I'd need to prove her age so I have no ID for her. Of course, I'd be prepared to have a birthdate ready so that if it even gets to that point, and they ask for proof, you say you don't have any, then they might ask what his birthdate is, if you have to think about it and make one up then you'll look suspicious.
Also, sorta OT, but Disney's policy that your kid is whatever age for the trip that he was when you began your stay, is that for onsite guests only? I mean, if we're in WDW for 6 days and my daughter turns 3 on day 3 and I say "she turned 3 yesterday but we started our trip 4 days ago so she's free" but I'm not staying on property, how do they know that I'm telling the truth? I guess they could send me to guest services and have my ticket looked up to see what days I used it on, if that's even possible, but would they go through all that? I would think no. So in that case, then maybe if I were trying to sneak in a 3yo I'd just say "today's his birthday" if it were after my check-in day and then he'd still be free.
With that said, we took my neice when she was 2 years, 9 months and was around 38 or 39 inches and about the same in pounds and no one once questioned her age.