The main problems facing arena mode are:
1) Bugs. During Invasion, a lot of people reported getting stuck in rounds that refused to ever begin. I don't think this got fixed.
2) Player count. Early on they decided to restrict the mode a smaller player count, and a lot of the maps are designed around that notion. As a result, sticking it in rotation with maps of other game modes is going to cause issues since a fair number of people have to sit out every round. On top of that, TF2's sort of drop-in/drop-out pub play doesn't match up with arena's short format (similar to CS), and an odd number of players means sitting someone out.
3) It's highly divisive. A lot of people hate it, a lot of people love it. Around the time that Valve was probably thinking about arena again (i.e. Invasion) Arena: Respawn was in its heyday, so they probably didn't see a reason to revive it when the people who actually like arena already had an outlet. I could be totally wrong about this though.
When Byre was added to the game, I don't think it solved any of these issues. Obviously it couldn't solve 1 or 2 without Valve's help, but it also
didn't exactly receive universal praise (though I don't think that was the case for
any of the Invasion maps), so whatever reasons Valve had for not supporting arena mode weren't really overturned.
The way that they ran the Invasion maps on rotation was a limited time deal, so any problems that might come from running an arena map weren't going to be around for very long. My guess is that they wanted to run Byre for their own reasons (because the map is really good of course!) and that any repercussions (buggy servers, sending false signals about reviving arena mode) were either unexpected or not important to them.