Never even noticed that option, though I have seen a few Amazon Prime branded delivery vans. If they're waiving the $2.99 for orders over $35, they should say so, it might incentivize some people to order more or combine items into one bigger order.
Considering how slow prime shipping has been (still takes quite a while for some items), that's just piggish on Amazon's part.
Likely not all cities have it, and of course not all Prime items are eligible.
Search filter available online (also available on mobile app, just harder to get to):
And how it shows by a product:
I've used it to get items "overnight, between 4AM - 8AM".
Seems like a hybrid between the regular Prime delivery vans (Amazon shipping), and Prime Now.