Well....
It COULD have been done. But, hindsight is 20/20, and there are LOTS of reasons why it took as long as it did.
1) Variolation WAS used in places a lot earlier than vaccination. (I.e. introducing a tiny amount of full virulence smallpox under controlled conditions, vs using cowpox). The problem with that is twofold, it didn't get to Europe by the time you want, and secondly that while death rates were much lower than in uncontrolled epidemics they were still a couple of percent.
2) 'vaccination' (deliberate inoculation with an Xpox orthovirus) was certainly doable earlier, under the right conditions. Do remember, however, that diagnosis was very iffy - and if someone had 'pox' and you used them as a source for further inoculations (you need to keep a reservoir of the active disease around), you might be spreading syphilis, chickenpox, etc., and making people very sick WITHOUT protecting them from smallpox.
3) 'cowpox' was epidemic, not endemic. It didn't exist at all times and in all places. So. You can't start this vaccination program unless the local cows have cowpox. (Hence, the reservoir note above, you have to keep a continuous chain of infections going to preserve the right virus).
4) related to 2 and 3, in addition to lots of other 'poxes' in humans, there are multiple 'poxes' in animals. If you use a 'pox of cows' that isn't an orthovirus, you won't actually get any protection.
5) there are LOTS of old wives tales. Most of which are false. Sorting out digitalis and vaccinium from thousands of useless nostrums is a non-trivial task.
6) if you inoculate a kid or two, and they end up not getting small pox, was that because they'd already had something, because they were lucky, because they were naturally immune, because no smallpox epidemic hit your village, or because the inoculation actually worked. The whole idea of scientific trials is MUCH later than you want, and ad hoc experiments are often annecdotal, and don't give 'proof'.
6a) if you inoculate those same kids, and they DO end up getting smallpox, did you have the wrong cowpox, the wrong human pox, they were immunosuppressed because of malnutrition, or because the concept doesn't work.
7) if you DO find the right cowpox, and you DO inoculate properly, and the kids ARE protected from smallpox, but then they get chickenpox or syphilis, then that will 'disprove' your cure because they still got 'pox'. (Again, if this is early enough that the various 'poxes' weren't understood to be distinct.)
8) suppose you hit the jackpot, and get established inoculation programs around Europe so that at least the wealthy are, indeed, protected from smallpox. (Who's going to pay to extend that to poor peasants/sailors?) Let's even suppose that vaccination (inoculation with small pox) is introduced to the New World with the first settlers, (<Sarcasm>because, of course, humanitarian concern is exactly why Europeans went to the New World, right?</sarcasm>). Then what? Not much.
a) cowpox is probably going to spread among the immunilogically naive natives, and kill a number of them (not nearly as many as smallpox, but still).
b) while small pox may have been the number 1 killer, measles, mumps, the common flu, and various other European ailments added up to a larger death toll cumulatively than smallpox. So you save some native lives, but don't significantly affect the dynamic of massive die-offs.