As long as it stacks, Refresh will still be asked for if there's a RDM in the party.
Furthermore, enhancements received through this ability can be used in conjunction with those received via magic spells, songs, the Phantom Roll ability, and so forth.
...should be the case.
Also keep in mind that SMNs can only do one effect, and will either compromise on other benefits, or the effect will not be full-time. The wording on it seems to indicate that offensive aspects of avatars may be nerfed when their JA is active. Even with that aside, using another avatar for anything will mean resetting the effectiveness after a period of not having it on anyway. Plus, figure, SMNs can't rest with an avatar out, so at times they may need to take a knee and, poof, no more effect for a bit. It'll also mean party members need to be within a certain range.
Compared to that, Refresh is pretty reliable. Really, Refresh remains the most dependable, least intrusive party MP recovery buff. Unlike Evoker's Roll, it's always 3MP/tick. It doesn't mean the RDM can't cast a different buff on you like Ballad(s) do. You (or the RDM) can wander off wherever you please, or stand where you need to, and it's still there. Just recast ever 2.5 minutes.
Given all that, I really don't see Refresh going away. If your party lacks a RDM, sure, there are going to be more options than ever for MP recovery buffs...but if a RDM is there, why wouldn't
you want 'fresh (outside of Sublimation)?