I think it's more a case of he was never mentioned in the same terms as May and Compton - I bet if we had an "England players draft" Barrington would get picked before them.
Besides the issue of stylish batting, you can see why the older England fans in the 1980s (with a bias towards remembering Ashes Tests) would have happier memories of May and Compton, who played in the Ashes wins in 1953, 54/5, 56, than of Barrington, who played in the 5 series in the 60s, none of which England won. Although Barrington was clearly England's top batsman in those series, and has a far better average against Australia than May or Compton, you couldn't say "remember that great innings by Ken which won/retained us the Ashes?" He didn't even play a notable innings in any of the Tests which England won.
Richie Benaud called May not merely the greatest English batsman to emerge since the war but the only great one
Was Peter May England’s greatest post-war batsman? Plenty of good judges thought so, as was reflected in his Wisden obituary in 1994. Read the tribute.
Barrington, like Boycott was a defensive batsman who was more likely to save a Test match than win one.
May and Compton on-the-other-hand looked to dominate the bowling and take charge of the Test match; and that‘s why the likes of Richie Benaud rate them so highly.
Personally, I play Peter May at number 4 in my England ATG team and let Barrington, Cowdrey and Root ‘fight’ for the number 5 spot.