I mixed up my playing style a bit, and came to an interesting realization: Garrisons are really enjoyable as "winding down" content.
Basically, you log in and do whatever your main goal is. Join a raid, do a heroic dungeon, hit up Tanaan, whatever. You defer the Garrison stuff to the end of the play session. After you're done your main content and are thinking about logging out, you spend the last 5 or 10 minutes taking care of your Garrison.
I find this works really well. Garrisons don't take much effort, and there's a nice sense of tidying up before you log. It's perfect for the end of a play session. You log out while seeing your followers off on their adventures. As well, being at the end, I find myself more willing to ignore the parts of the garrison which are completely unnecessary. Like the herb garden, since I have no need for herbs.
A player named Torvald had an excellent post where he postulated that doing garrison stuff up front drains a player's "stamina" and enthusiasm for other content. I find that simply moving the garrison to the end of the play session eliminates that entirely. Maybe it still drains your stamina, but you were planning to log off anyways.
There are a couple problems with this style of play though. First, it's sub-optimal. Since missions are timed, you lose out on the time during your play session. Second, it requires you to defer some easy gratification, some easy rewards. And as we all know from various experiments involving children and marshmallows, deferred gratification is not an easy thing.
But you're still faced with the deferred gratification problem. The marshmallow dangling in front of your face when you first log in. You can't put a timer on it, because the player might be logging in quickly just to do garrison tasks.
The only idea I had was that while your followers are hanging around your garrison, they give you a stacking buff which increases gold, xp, and valor gained by 1-2% per follower, up to 50% when you have a full complement of 25. That way you want to delay sending your followers on missions while you are doing content, but you're perfectly happy to send them off while you are logged off.
In conclusion, I think that garrisons are much more enjoyable when moved to the end of a play session. But I think actually nudging players into that playstyle will be hard sell.