It seems that the name for Kirk's ship was picked more or less by random, perhaps mostly in honor of the WWII aircraft carrier (in the real world, the most "heroic" of the lot), at a time when Archer's voyages had already been forgotten - and it was Kirk's ship rather than Archer's that made the name Enterprise a prominent one in Starfleet.
I'd have to agree with this. It seems that the NX class vessels were named after the Space Shuttles, while Kirk's Enterprise was named primarily after the U.S. Navy aircraft carrier, as other ships in TOS were primarily named after naval vessels. There doesn't have to be any correlation at all between Archer's ship and Kirk's ship as far as the nomenclature goes.
That's because there was no space shuttle Enterprise when TOS wae made. And of course, the shuttle was named for Trek's ship, and first appeared in-universe in TMP, as part of the rec room display.
I wonder what Trek's version of Enterprise OV-101 was named for? I guess that goes in the "universe exploding questions" box along with what the Trekverse lyrics are to the Beastie Boys' Intergalactic