You'll forgive me, but often changing keys is nothing more than playing the same fingerings in a different position on the neck. Of course, right down at the nut (the classic 'tuned down to E-flat so I could play it') things may have to be re-arranged, but in a lot of cases you just move up or down a fret or two or three. I will say I find it far easier on a five-string, where it seems for me it's harder to run out of room down at the deep end vs. a four-string.
I too have rarely been advised to learn something in every / any key, and would think that applies more to instrumental music, as I've rarely played with singers that could get very far from their native range.
It's not nearly as hard on a guitar-style instrument: It will REALLY grow you a pair on keyboards, where there is a huge difference because of the black keys. Believe me, the same song in C Major is W-A-Y different and harder to handle transposed just a half-step to C-Sharp Major. Doing that on a necked instrument is a breeze alongside of that.
Sure, it's a hoot to blast along on open E's and A's on more simple music, but just as often you can forget about that, there's always going to be songs in F, G, B-flat, and on and on.
In Nashville, where we use the Number System quite a bit, you learn to almost quit caring in terms of what key a particular song is set in: Someone will call out a tune and a key, and you're off. You think of it as where to be on the fingerboard to start from, then walk through the changes in your head as numbers instead of chords. In this way, it's a great way to break yourself of not wanting to play in certain keys and think in terms of intervals and where to be on the fingerboard.