It's not really about Barkov though. He doesn't come out terribly per 60. It's just that Matthews has elite rates. There's no amount of usage that is going to put Barkov where Matthews is, right behind McDavid and MacKinnon. He's just not at that level of offensive talent.
Basically every advanced stat model and website uses per 60, so it's not going to go away despite any changes in Matthews performance. And every fanbase has hypocrites, this is how debates here work.
The problem with these per 60 stats is it's too prone to anomalies that actual production wouldn't have.
For instance, if one were to assume that the best ten forwards in hockey in 2016-17 were based on points (obviously a gross simplification, but just for this example), which method paints a more accurate picture?
Top 10 actual points at 5 on 5:
Connor McDavid - 63
Patrick Kane - 52
Mark Scheifele - 52
Sidney Crosby - 50
Henrik Zetterberg - 49
Artemi Panarin - 45
Brad Marchand - 44
Nick Backstrom - 44
Jason Zucker - 44
Nikolaj Ehlers - 44
Top 10 points per 60 at 5 on 5 (min 800 minutes played):
Conor Sheary - 3.02
Connor McDavid - 2.87
Evgeni Malkin - 2.82
Sidney Crosby - 2.68
Mark Scheifele - 2.55
Scott Hartnell - 2.42
Arvidsson - 2.42
Marchand - 2.39
Backstrom - 2.38
Pominville - 2.38
The bolded are guys who have absolutely no place in any sort of "top ten forwards" lists. The actual scoring list looks more representative of what you'd expect if you asked people to list their top ten forwards than the per 60 list.
That's my biggest issue with these per 60 stats. You have situations where someone like Conor Sheary has a fluke year in minimal minutes to skew the results, whereas that doesn't happen with actual production.