I don't know there is a good answer to these questions. I am not a big fan of monster PCs because I feel that most of them don't fit in well with a standard D&D group from a story point of view and most of them have abilities that are significantly more powerful than humans/elves/dwarves/etc.
From what I'm reading elves, dwarves, (and presumably) halflings, orcs, gnolls, etc. should all be PC races, because that is fantasy enough, but "monsters" aren't? Are they just that stretch just too far? Where do we draw that line? Elves are good, minotaurs bad. What about centaurs? Why are ogres too far, but orcs are okay?
As I said before, I've had monster-PCs before. They are more work but they are a lot of fun. There are certainly problems but I don't think those problems can't be overcome. Maybe Dausuul is right and multiclassing is the problem. I don't exactly know. I think PF did a great job because any monster could be used as a PC and because ones with huge CRs (level adjusts) could be bought off over time, making an ogre PC eventually (possibly) end up with as many class levels as the human PC.
As hard as it is to balance a wizard vs a fighter, it's even harder to balance a fighter vs a dragon or even a Mind Flayer. No ability you could ever give the fighter would really be as powerful as an area of effect stun for a long period of time like a Mind Flayer's. It doesn't matter how many level adjustments you give it.
This is hard because (a)why are we trying to balance them at all, and, (b)what level are you using as a starting point.
For A: As you said, fighter-wizard dichotomy is hard enough. But by in large 5e isn't trying to balance them. Being off balanced or over powered is a real concern they are working on but as the article says; an ogre PC is going to have certain advantages, like massive strength and power, but huge disadvantages in social situations. That seems like a form of balance to me. Using weaker forms (though I have unique issues with that in itself) is a way to do this, where an outsider PC perhaps doesn't get teleport-at-will. That is balance. It is hard but attainable just as having someone with teleport, or invisibility, or flight is hard, but not impossible, to balance against someone who is mundane and lacks those tricks.
For B: I haven't seen anyone, who wants monster PCs, say they want to play a dragon at first level. Nor a mind flayer. In fact the few people I have seen post about that says (myself included) that they would be willing to wait to get a full blown version. Again, the weaker monster could be implemented here. Or a buyoff as I (via PF) suggested. Heck even my half-baked Half-monster idea could work. In fact no one is trying to balance a dragon vs. a first level PC. An full grown adult ogre (aka ogre classic) in a first level party won't work either. Having to wait until the party reaches.. 3rd I think they said, before able to take the ogre as a race seems very reasonable to me. The problem after that seems to be based on advancement past 3rd level, where the monster isn't getting much out of their adjustment.
Oh, unrelated to those two above, Level Adjustment:
What if LA wasn't a number added onto the HD. What if the monster had HD (along with HP, saves, and abilities) and that was it. Some of those HD might be relatively useless and you could even have a reverse LA, wherein you reduce the effective level to UNDER the HD total. That reduces the problem of saves (though that is much less an issue in 5e) and HP with a HD 4, LA 2 monster who ends up with far less HP than a 6th level PC. Just a thought.