I have no real experience with 5e, and with respect to combat I don't have a huge problem with the bounded accuracy mechanism and the smaller gaps in power. I think that what you are discussing was the intended goal of the bounded accuracy mechanism, and I understand what they were going for and why.
My problem with 5e, and one of the reasons I didn't adopt it, is something else that is in the implications of this statement:
"With bounded accuracy, and rare special expertise bonus nonwithstanding, a 20th level character should have 20 in his main ability and +6 proficiency bonus, so he'll roll at +11. Let's add a legendary item and it might be +14. But even on a 20, he won't be much above what a quite gifted but otherwise regular joe (+2 proficiency, +3 stat bonus, level 1) can sometimes do (max 34 against max 25)"
This range makes sense within the bounds of combat. What it doesn't make the slightest bit of sense (to me) with respect to is non-combat challenges. 3e introduced a very solid non-combat skill system, albeit one it was cautious with respect to and didn't push as central to play, but that solid non-combat skill system was one of the things that brought me back to D&D and something I consider essential to any modern rules set.
The problem with the 5e approach isn't just legendary skill in task completion isn't that much greater than everyman skill task completion, it's that neither legendary skill nor everyman skill is sufficient skill to be reliable at a task. Now, from a D&D perspective, not being reliable in the completion of the 'to hit' task in combat is a good thing, since D&D simplifies away active defense into an abstract but fast combat system. And D&D has other levers like hit points and damage to differentiate legendary skill in combat tasks from everyman skill (and everyman skill from unskilled). So none of that is a problem with combat, but it is a huge problem with non-combat tasks where reliability is both desirable for world building and often a desirable hidden class feature.
To understand what I mean by that, there are by my way of thinking two sorts of skill uses - reactive (or passive) and proactive (or active) uses. Reactive uses of a skill occur when within the story or scenario there is some non-combat challenge which requires a skill check. A good example of these is the common use of perception or lore checks in most systems (whether D20 or BRP) where if the PC has the right lore or notices the clue, they receive from the GM some intelligence about the story or scenario that is supposed to inform their choices and lead to better outcomes. Critically, though, the utility of a skill in this case - high though it may be - always really depends on the GM preparing a particular clue (either before or on the fly), and as such really is scenario dependent. Your 'Knowledge (Religion)' or 'Ancient Greek' skill is only going to be useful if the GM has crafted a scenario where that skill is useful.
But other skills open up choices for your character that you didn't have before. One of the most obvious sort is athletic skills that let you declare movement that you couldn't declare otherwise - climbing, swimming, jumping, tumbling, or whatever. These are 'proactive' skills, where quite often you can attempt things that the scenario doesn't explicitly plan for, and where the player initiates the skill usage by proposing some action. In my experience, the critical thing about a proactive skill that makes them really useful is that they achieve some level of reliability - you can always clear at least this much distance, you can always balance on a ledge of this sort, you can always climb walls of this sort, and so forth. If they don't become reliable, then really they tend to remain reactive - mere savings throws to be made to avoid some GM initiated trouble that is coming the party's way. If they aren't reliable, then the risks of the relying on them - of accepting more trouble - don't outweigh the benefits, and the player only uses them when they absolutely have to.
If you look at what can be faced reliably, the DC's are really quite low. Sooner or later, anyone is going to fail that DC 15 check. And yet a DC 15 represents something that even an unskilled person can do 25% of the time.
And to me it feels like inadvertently or deliberately, the 5e skill system was designed with the idea that skill checks are just a sort of saving throw in mind, and as such reliability wasn't a goal of the system.