In D&D 5e roles overlap a lot. A ranger can be a capable healer, a fighter or a cleric could disable traps, druids are excellent scouts, bards could... well could be anything from healing to diplomatic duties to thievery, etc.
Sure, but they don't do those things very well. That's the problem you get when everyone can do (a little of) everything. Nobody does anything particularly well.
Nah, that was in previous versions of D&D. Bards could master any skill, only some rogues could rival them. They also could fulfill any role you can think of if you build them accordingly.
Rangers have healing spirit, something that even clerics do not have, so they are fantastic healers. They only lack mass healing. Even celestial warlocks, bards or some sorcerers have buff and healing capabilities.
Druids have pass without a trace and could turn into animals ( even animals with swimming, flying or burrowing capacities later on), and also have several spells at their disposal to detect enemies. They are one of the top scout class of the game by far. Wizards and Bards excel at it using their familiar to scout ahead. There are several hybrid subclasses that allow fighters or rogues to have access to wizard spells, so they could provide arcane support.
Any character with enough dexterity could use thieving tools to open locks/disable traps, etc. And since there are weapons that use dex for attack and damage, there are fighters, monks or rangers that could be excellent trapspringers.
Paladins, bards or sorcerers could be great faces for the group, due to his high carisma, but any character could learn the dialogue skills. etc etc
I don't buy this. But for the sake of argument, if you are right, then what exactly is the point of having classes? Effectively, every class is considerably the same as at least one other class and maybe more, and all the classes significantly overlap each other. So in truth you have a classless system with fake classes to give the illusion of a class-based system. Nope. Don't buy it at all. A cornerstone of D&D has always been its distinctive class-based system, and although 5e has weakened the distinctiveness of classes it has not erased their distinctiveness the way you describe it.