Bairstow always said he wanted to keep even when first selected to bat. That's a big difference to ABdeV and Sangakara he were forced to keep and were lessened as batsmen. There is a big psychological difference. The biggest argument against him keeping was he used to be below average. He's now a decent keeper.
As a batsman he'd struggle above 5 because the bottom hand still closes the face of the bat. He's averaged 40 plus since keeping. I don't think he's ever going to be in the very top tier as a batsman whether he gives up the gloves or not. Maybe an extra couple of runs an innings. At the moment he's had one stratospheric year, one average and three poor years. I suspect if he keeps the gloves he will end as one of our 5 best keepers, personally I don't think he'll ever quite make it as one of the top 25 batsmen.