Even D&D 4e was dramatically more popular in terms of raw numbers than virtually anything else on the market. So even though a presumably significant part of the base recoiled from it, the amount left was still enough to largely dwarf everything else.
That says there are issue going on here well beyond anything to do with system design.
So I think very few people are going to have an issue with the notion that 5e's design has a meaningful impact on its popularity. I think almost all the pushback comes from the idea that a more popular game is necessarily a better game.
I think the comparison to 4e is interesting.
Back when 4e launched, some people would quote sales figures which at least purported to show that the early 4e books sold more copies than 3E books in the same sort of time frame. Is this evidence that 4e was
better designed than 3E? (My view: no.) Is it relevant to how PF 1E fans engage with their preferred RPG? (My view: no.)
What happens if, one day, WotC launches 6E and it is
even more popular than 5e? Would that prove that 5E was flawed in its design? (My view: no.)
Here's a conjecture that
could be true, based on my knowledge of the relevant evidence:
* 4e made it possible to build a certain sort of market presence - eg drop-in organised play; streaming of play - that 3E didn't support so well;
* 4e had design elements that put a relatively hard cap on how much that market presence could grow;
* 5e builds on that established market presence but overcomes some of those hard limits.
A more general version of the above conjecture:
4e is a necessary component of the development path that made 5e and its popularity possible.
Now I can't prove these conjectures. Maybe there was a possible pathway to 5e - with its fixed damage expressions for spells, its short rest/long rest structure (which I gather of late is being abandoned), its death/dying rules, its monsters and NPC built on a different framework from PCs but still having the same build elements (stats, skills, powers, etc), etc - straight from 3E without going via the 4e versions of those things.
But suppose my conjectures, or something in their neighbourhood, is true. What does that tell us about 4e? And what does that tell us about 5e's eclipsing of 4e? I think it becomes something more complex than simply that 4e failed and 5e succeeded.
All that said, such conjectures and conclusions seem to me to be completely irrelevant from the point of view of someone who wants to play and enjoy 4e D&D. The fact that the game has hard caps on how much its market presence can grow is super-important for a commercial publisher like WotC, but utterly irrelevant to those actually playing the game.