Robert April clashes with Enterprise because Robert April was the first Captain of the Enterprise. Not some twit named Jonathan Archer. If they'd used April and Sarah Poole in the Enterprise series (with or without the dog; I liked Porthos), I wouldn't have complained half as much. I'd still have complained about the stupid stories, but not about who the Captain was.
Note that "The Counter-Clock Incident" explicitly said that Sarah April was "...the first medical officer aboard a ship equipped with warp drive" in reference to the Enterprise, completely ignoring the starship Bonaventure from the earlier TAS episode "The Time Trap" let alone the TOS episode "Metamorphosis" which had previously put the invention of warp drive much much earlier, and think about how stange it is to get nitpicky about the name Johnathan Archer on what was clearly supposed to be a different starship Enterprise (they don't even look alike) that even flew under a different flag to Kirk's version (Earth instead of the UFP) a whole century earlier.