The thing about the Lyth/Bairstow 'earning' selection through sheer weight of runs is that it really shouldn't matter (at least not on its own).
If you aren't good enough to play Test cricket, or you aren't in the best 6 batsmen in the country, then no amount of FC runs should mean that you have to be picked. England didn't have to pick Mark Ramprakash from 2006-2009 when he was averaging 100 every year, because he was a tried-and-tested failure at Test level and nothing had changed in his game to suggest he'd succeed if he got yet another chance. If you don't do the job at Test level and you keep racking up FC runs at a faster rate than anyone else, you haven't earned a spot.
In the case of England here, it's very much understandable picking on form -- Lyth made sense not just because of his weight of runs, but because there wasn't an established opener. Lyth was no more nor no less likely than any of the other non-Compton opening options to be of international standard in the medium term, so picking the most in-form of those options makes complete sense. Lyth was the most likely of them all to make jammy 30s and maybe the odd attractive 50. Didn't quite work out well, but hey, the selectorial logic was sound enough. Apart from not picking Compton, that is.
The case of Bairstow is a bit more difficult -- he's very much still in that "is he or isn't he Test quality" stage, with selectors balancing whether his dominant FC season was representative of him changing compared to the out-of-his-depth Bairstow from 2 years ago, or of out-of-his-depth Bairstow being too good for FC but not good enough for Tests. Were the runs a result of him fixing his flaws that had him found out at Test level last time, or were they in spite of them? Were his Test failures because he's in that awkward 'in between' spot, or was he just in bad form? And, if you think he has changed and is good enough, how much leeway does he have to settle in -- how much starting credit does he have in the bank?
If Bairstow still doesn't perform to the required standard by the end of this winter, and he starts off the English summer with 1000 by the end of May (i.e. nothing wrong with his form, he's just delivering at FC level but not at Test level), do you keep picking him anyway because he's 'earned it' in the hope this time will be different?
If you go through those questions and come out deciding to pick Bairstow, that's fine. This isn't an easy question to answer, and giving up on a promising talent prematurely is just as bad as showing too much faith in that potential. But let's not pretend that Bairstow is owed that spot by virtue of his FC runscoring. Sure, he's done everything that could possibly be asked of him, but that, in itself, doesn't make him good enough to play for England.
(ftr replace 'Bairstow' with 'Shaun Marsh' for the Australian example; I don't mean to attack Jonny personally -- this is a broader point re. selection)