Bairstow is better than Burns or Hameed. I think they are getting far too preoccupied with opening batsman. It should be about who will score most, not who is experienced at 1 & 2 in the order. If someone will score more runs then tell them they are opening the batting and to look to rotate the strike and be proactive.
I honestly don't think Bairstow could open in Australia, regardless of how many runs he was scoring down the order. I think out of this batch I would say only Root has what it takes to open but he is definitely better at 4, especially with captaining.
Some batsmen prefer different positions for a reason, as there are different skill sets, i.e. you wouldn't necessarily have a blocker playing at 5 or 6. Root clearly has a better record at 4 than 3, which is a strange anomaly as he is usually in just as quick. It is a mental thing, look at all the best openers they all had 2 things, they were technically sound but they also had mental strength, not sure there are too many English batsmen that fall under both categories in a strong way and that is our problem.
Also, pretty sure Bairstow has been tried at the top and failed, I might be wrong though.