Big caveat: I'm not a lawyer, and you absolutely should consult with one.
I'd be very concerned about missing key rights and responsibilities by not being included in the LLC agreement. Beyond that, as long as your partner is treating it as a single-member LLC (and assuming you'll have losses at the beginning) he'll be getting tax losses that should be accruing to you. Most importantly, why the reluctance to do this? If not now, when?
In short, I agree that it changes the structure of the LLC. But that's to be expected, can be dealt with easily, and is exactly what he's signed up for by bringing on a partner. If he's reluctant to make the company's legal documents reflect reality, I'd treat that as a pretty big red flag.