I think most UK fans would have been okay with UF, UT or Vandy. Not so much if it had been a team from the remainder of the conference.
As far as rivalries the UL, IU and UNC games are pretty much it. A multi-year deal is in talks with Duke as well. SEC games are just that - conference games. Some obviously bigger than others but not the same as the OOC rivals.
If they go to 16 teams there is a pretty good chance they would go to a 4 division format. UT, UK, Vandy, plus 1 more team (dependent on who the 2 expansion teams are) would probably be the North division. that would have UT back to playing UK twice a year again.
Tiered Rotation is the way to schedule and will give UT the opportunity to play the best conference teams home-and-home each year. Here's how it works in the Big East:
Big East ponders conference scheduling with eye toward incoming teams - college basketball - ESPN
Look at how many teams the Big East puts into the Tournament due in large part to the RPI advantage of "the-best-playing-the-best".
Here's how the-best-playing-the-best tiered rotation would likely work this coming year for UT.
Home-and-Home - 5 Total
Fixed Rival (1): Vandy
Best Playing Best Home and Home (4): Kentucky, Florida, Missouri, Alabama
New SEC Basketball Schedule: How a Tiered Rotation Might Work - Rocky Top Talk