Although the success rate is very close if not the same with each set of weapons,...
I've never seen any official stats on this but I'd bet my house it isn't true on average.
For example, I have a friend who's been in competitive archery since the 1970s. First he shot long bows and recurves then the same with "clickers" then he went to compounds and fingers with "clickers" then compounds and back-tension releases. He's been a national level competitor for 40 years.
His description of long bow hunting is "Most of the time, we were thrilled to have an arrow with hair on it, say nothing of making a kill, say nothing of a "clean" kill. If somebody killed one, it was practically a celebration of a life event."
Now, I have no doubt (none) that there are places where there are so many more deer than there were here, then, that the traditional archers get a fair number of deer. However, just the mere fact that the reasonable ranges are at least double and quite probably triple or more tells me that the success rates can not possibly be remotely close.
Look at it like this. If you have an effective range of 15 yards, you've got 707sq.yards in your effective range. If you have an effective range of 30 yards, you've got 2,827sq.yards in your effective range. All else being equal, just by those numbers alone, you're going to have 4 times as many deer in range. Add in the indisputable fact that an average shooter of both disciplines will shoot much smaller groups at any distance with a compound than a traditional bow and the success numbers go up farther. Even if it's 20 versus 30, you've got more than twice as many deer in range.
My friend is a perfect example. He once told me that he took "a few" deer with a traditional bow. I'll have to ask him the exact number. He just last year made it his 25th consecutive year with
at least one deer with his compound. 2 years ago he got 3. I won't be the least surprised if those 3 were as many as he got
total with a traditional bow. Remember, he was a national level competitor and hard-core hunter with a traditional bow.