It's not so much that there was an Enterprise before Kirk's (after all, the TMP rec room scene featured an earlier iteration--the "ringship"/starliner version), it's the concept--one of Earth's most important and famous ships was called Enterprise. Before Kirk's. And Archer is said to be one of the greatest explorers of his time, later becoming Federation President.
Because ENTERPRISE--the show and the ship-- are IMPORTANT, don't ya see?!?!
A long-lost notion of TREK was that Kirk's Enterprise was just another ship which eventually became distinguished, but was not at all the "flagship of the fleet". Famous and historic? Sure. But not the end-all, be-all of the fleet. Throwing in a prior Enterprise which did all of these great and important things retroactively makes Kirk's the only one in the franchise that wasn't IMPORTANT from the start.
The deification of the Enterprise and her crew is a classic example of fan-think creeping into the writing, and writers having the characters know or believe things that only viewers should be privy to.
Just like fans asking why people don't figure out Superman's secret identity. Except...why would Superman even LET ON that he has another identity? Sure, WE know he has one, but why would anyone else in the DC universe know that?
Anyhoo, the NX-01 stole a number of "firsts" from TOS, and other iterations, too--a katra transfer with a human, first meetings with the Organians, Borg, Ferengi, genetically-engineered supermen from the 20th century, etc. To say nothing of goofy retcons, like female Vulcans going through Pon Farr.
All in all, ENTERPRISE suffers from massive doses of "prequelitis", and does everything in its power to overwrite or sidestep TOS. The look and feel of the show brings to mind a TNG-era prequel, more than TOS.
We also see "phase pistols", the first real starship (NX-01, which can be taken as the very first "NCC"-type ship), the first use of a transporter aboard a ship, first contact with Romulans, Klingons, etc. All of these historical and technological events are conveniently centered around one ship called Enterprise, which was incredibly famous and important long before Jim Kirk's ship. And, of course, Kirk's ship would now have a legacy to uphold, as the second starship in the series, rather than being just another ship which eventually stood out on its own merits, thanks to the events of TOS.
Granted, this level of coincidence isn't anywhere close to Abrams' film, but...
Really, a casual viewer might very well watch ENTERPRISE and then move on to TNG, without even realizing that TOS is supposed to fit in-between.
And, now, we have Abrams' TREK to replace TOS. In fact, that film does a halfway decent job of filling the gap between ENT and TNG (aside from certain elements, like the destruction of Vulcan).