He hasn't posted back in a week (maybe he's just lurking)...so we don't know if he's tried anything different, solved his problem, or just moved on and forgotten about this thread.
I use to hate the SM57 for similar reasons...always came off harsh to my ears...and if you moved of the center too much, you lost the bite.
Anyway...after moving to a Cascade Fat Head, I initially loved it, but then started to feel like it was too dark...so then I tried the AKG D1000E, which sat right in-between the two....and I went with that for a lot of stuff.
I then got a new amp, and didn't like either the Fat Head or D1000E...they weren't giving me the crunchy Brit tone I was after...and I came full circle and back the SM57.
So there is no one "tone"...and IMO, there is no one mic that works on everything. Granted, if you dial in a singular tone that you use over and over because that's the style of music you are doing and that's the tone for it...then you use the same amp(s), the same mic, the same position, and it's a no brainer....but if you're doing lots of variety...you have to try different things.
I use all three mics now...depending on what tone I'm after, since the each have a different flavored sweet spot.
AFA what the OP is after...the White Stripes sound (or the stuff about Glyn Johns)...you can't assume anything about what they used and how they did it.
They may have used an SM57...but there are a bunch of other variables that would figure into the equation and the final sound...so basically, if you want to nail it like they did, you have to almost replicate all those variables in your home studio...which can be difficult if you don't have all the main ingredients that they had. I mean...it's not just the mic.