When I see 15+ rolls (with all bonuses applied) fail, and realize that that means 50% or more of the possible rolls would have failed, I begin to think that yes, it is somewhat representative of what's going on. Rolling 10-or-less happens 50% of the time, after all. I understand probability and statistics at least at a basic level, having taken a year's worth of it (and done quite well, I might add).
Edit:
In fact, rereading your post, it occurs to me that that may be exactly the problem. I don't think someone "Proficient" with something should have equal odds of failure vs. success on basic tasks. I think someone who barely knows what they're doing, but has a competent natural talent (read: good ability score) should have that. Someone actually trained--especially if they also have natural talent--should be doing a hell of a lot better than 50-50 odds of success.
While that is possible, I find it unlikely to apply to my case, as the guy DMing for my 5e group has never DMed before. (Technically that's not completely accurate; he had begun DMing a different 5e game a few weeks before he started ours. But he's never, to my knowledge, DMed any other edition or even any other game before 5e.)