Buttler decided to concentrate on the white ball stuff (good word) years ago. He stopped playing First Class Cricket almost completely and didn't anticipate being a regular in the Test team. If there were half decent other options he wouldn't be playing.
More complicated than that that’s for sure. Huge oversimplification.
Bairstow averages 27.41 as a specialist test bat, but 37.83 when the nominated keeper.
There can't be too many players with such a marked difference with reasonable sample sizes, surely?
Intuitively one would imagine most average more without the mitts.
For comparison Buttler averages 31.65 with the gloves, and 35.68 without, a slight but not negligible difference, which one suspects is the case with most others, the McCullums, Sangakkaras and Walcotts of the world.
Seems as if we've given the wrong keeper the gloves all this time. Although, obviously, other factors (like moving Jonny up the order when he's not keeping) have played into this.
The comparison doesn’t work as well as you think, you need to think where they are batting. Well, you did touch on it - but the point is it’s practically the only reason for the contrast.
Neither of them are capable top 5 batsmen in tests. When Bairstow doesn’t keep he almost always bat top 5. So he’s averaging 27.41 as a specialist test bat yes, but he averages 37.18 at 6 and 38.55 at 7 (38.33 at 8).
He averages 28 in the top 5 (52 innings). 37.86 at 6,7 and 8 combined.
Buttler’s batted top 5 in 7 tests (10 inns) and averages 23.3 here. He also averages only 29.7 at 7 and 28.25 at 8 - but 47 at 6 (17 tests / 24 innings -unfortunate for him he’s not been able to bat in his best position more, though sample size isn’t enormous it’s still reasonable and certainly enough to hold weight). Also to consider when comparing batting alone is that Bairstow with the gloves still averages more than Buttler without them.
Buttler’s performance at 6 it would have to be him for me if it’s just one and the keeper bats 6, room for Bairstow as a speciality bat at 7 if you want him. But if the keeper is batting 7 it’s Bairstow by a distance over Buttler! (What I carry on to say suggests Foakes batting at 7 looks a very strong option too, making room for Bairstow as a specialist bat at 6 - or if Buttler plays as a specialist bat at 6 you’ll have to hide him in the field.)
Who gets the gloves when both play - this is complicated by the fact Bairstow can field better. Buttler isn’t test match standard without the gloves in my opinion. Isn’t quick like Bairstow, athletic like Bairstow or a very strong arm like Bairstow. You can also try to put Bairstow in the slips where England don’t have many options. You can’t for Buttler - Buttler for example also dropped a lot of catches in a series without them - I mean a lot - he got the joint record (with Kamran Akmal) in the CricViz database (tests since 2006) for most dropped catches by an individual player in a series since they started recording and it was just a 3 test series (2019 series vs West Indies). That’s not to mention other games. Bairstow isn’t a flawless fielder, no one is. But clearly better than Buttler. The difference between them with the gloves isn’t much, Buttler has been keeping very well lately. Hiding Buttler in the field further complicates it, you’d have to consider the other players (how they can field) and which of the two other keepers to select. We can’t consider Foakes as a specialist bat - I suspect he won’t be a good fielder. We wouldn’t be considering Buttler if it wasn’t for his performance specifically at 6 - something he is highly unlikely to maintain long test considering his overall test stats and fc record (though not impossible) - but the fact he has to be hidden on the field, plus the low sample size (as mentioned), low fc record (as mentioned) and “low” overall test average I think it’s unlikely England would play him without the gloves again (it’s a possibility you could certainly try to justify though, 47 average at 6 is very strong)
Can also consider Foakes since it’s been mentioned, though this was just about their batting. Foakes is a good keeper, but in my opinion overly bigged up as a keeper - but the best of the 3 with the gloves. It should also be noted he has a decent fc batting average compared to Buttler, comfortably better - and noted that all 8 of his tests have been away from home and it’s a small sample size but he’s had to bat out of position at 8 (23.12) more than he’s batted at his strongest 7 (43) and it’s a big performance difference - (Bairstow and Buttler both average more at home like most players, you’d expect the same for Foakes).
There’s a case for all three keepers in tests at different times, the conditions to consider and where they are batting, the balance of the side - if Stokes can bat 5 (giving the potential option of selecting Bairstow as a specialist bat at 6 or 7 plus Buttler at 6 or Foakes at 7), If Stokes is injured then who are the alternatives and so on. But it’s obviously very hard (impossible in my opinion) to justify any of them in the top 5 with or without gloves by what I’ve said.
Think I’ve covered enough, potentially in a roundabout way as I mentioned Foakes but should make sense. This is without even discussing Bairstow vs Buttler in limited overs!