Well, Blizzard preaches that choice of covenant is up to players, but sadly we all know thats not how this game works.
So yes, by the looks of it - Covenant choice sadly depends on your main spec.
For instance, like i mentioned earlier, look at DKs "Swarming Mist" Covenant ability and then look rest of them. Swarming Mist looks like obvious choice in all cases. Or warriors "Conqueror’s Banner" ability for group play or "Condemn" for soloing. Conqueror’s Banner looks far too good for most cases again due to sheer utility.
They did say that you can switch Covenants, and progress with new one keeping the progress on old. However it will not be easy, nor cheap, as its not intended for us to swap Covenants frequently.
In addition, all covenant cosmetics are covenant locked so you wont be able to use covenant transmog that you earned if you are not specced into that covenant.
Honestly, Covenants should be scrapped in their current form.
My idea was that we should get another talent row with unlockable talents as we cap covenant progression - with 4th Covenant being class wide ability.
Picking and sticking a covenant would provide boost to talent that you unlocked via that covenant, with minor boots to other talents as well. mostly passive.
This way you would experience all zones and covenants, but also have a goal for your progression. In addition, not having "optimal" covenant wouldn't hurt as much as it is in current iteration.