I would read "marked on the product page and at checkout" as saying that products just happen to be marked in both places, not saying that it might be marked in only one place and thus not be eligible.

I accept that it happens, but if it does then they are outright lying when they talk about what they offer, and because they are offering two-day shipping then Prime definitely does have everything to do with it, even if they might try to back out of it sometimes.