In my opinion there a few ways of determining if the game has a problem.
Objective issues. For examples look at 4e's math. The math was off and that lead to a patch that involved a feat tax to correct PC bonuses. Both the original bad math and a feat tax to fix it were objective problems. As is CR in, well, every edition that has it. CR doesn't work and never has. There are too many variables between monster abilities, PC abilities, and player abilities for a CR number to be accurate except by coincidence.
Another way to see if the game itself has problems is to look at what it is trying to accomplish. If the game is trying to accomplish gothic horror, but all you can really do with it is Scary Movie, the game is having problems.
If someone comes here(or any other place) saying, "I have a problem with X, therefore X is a problem," they have already failed to prove the problem exists as a whole and needs to be fixed. They need to come here and leave themself out of it, showing just how the game objectively fails or fails to achieve its goal(s). Or they can say, "I have a problem with X, because it fails to achieve goal Y and here's why I'm saying that."
That's why I say "High level 5e is objectively broken because casters have too many spell slots and this reduces the number of playstyles that high levels can run to 2 because 1)the DM has to burn down a lot of spells slots in unrealistic ways in order to create a challenge and 2) in order to burn though all those spell slots, he DM must make the caster look at all their complex spells and lengthen their turn dramatically"
I said this over and over.
Then people say "High level is fine. I run it."
Then when you learn their player turns and monster turns take foooooooooooeeevvveeeeer and the DM is always metagaming way to sneak in 8 combats to burn through 16 spell slots on 2-3 casters each everydaaaaaaaaay
I say "Do you realize that very long turns and the need to metagame in filler fights on archmages and high priests all the time might cause people to not play high levels?"
"Well that's D&D. Go play some other game. It's annoying but I like it. You can't please everyone."