As mentioned, it can depend somewhat on the rabbit. I had one years ago that snuck sausage, salmon and chocolate and seemed no worse for it.
As for the high sugar in the carrots, it is actually worse if they don't have any for weeks and then suddenly get a lot. That causes a sudden surge in sugar levels which can cause a sudden growth of bacteria which can then upset their bacterial balance. Makes no difference if or when they were weaned onto veggies, it's just the potential upset of that bacterial balance in their gut which can fluctuate wildly (and dangerously) if suddenly given a bunch of sugar.
If they can handle higher sugar levels, it is safer for them to have it consistently rather than as feast and famine.