I'm no expert, but reading the bios for CV-6 it seems to me that CVN-65 and the upcoming CVN-80 are very much intended as the successors in every way to CV-6. However, as I'm not an expert, I'd be interested in hearing why you believe they aren't?
When CV-6 went out of service in 1946, the ship to fill the gap was the next of the larger Essex class carriers in line, technically the Valley Forge (CV-45). The more or less contemporary but more capable Midway carriers did not fill that gap, but met a different need. The next ship named Enterprise was not yet even a glimmer in the eyes of her would-be designers at that time and age - she entered service more than a decade later!
Similarly, when CVN-65 was retired, the carrier to take her place was Gerald Ford, CVN-78; one could also argue that her role had already been taken by George H.W. Bush, running up to operational status when CVN-65 was winding down. The next ship named Enterprise, CVN-80, does not even exist yet, and will not address the gap left by the retiring of CVN-65, but another gap entirely (real or perceived, arguably the approaching retirement of Nimitz or Dwight D. Eisenhower but alternatively a desire to at least temporarily increase overall carrier capacity).
Names are irrelevant to the issue of "succession". It's not as if USN would have been missing a ship had CVN-65 been named MacArthur or Roosevelt...
Timo Saloniemi