CapnZapp
Legend
I've run almost all of it by now (my players' characters are level 14), and I can say using it as written has worked well. That is, they level up approximately once per chapter.I'm reading through Out of the Abyss now, and it's kind of an odd adventure structure. The adventure expects PCs to gain a level per chapter, but the chapters are very free-form and most don't contain nearly enough challenges to gain a whole level, even if you do all the things in them. I think they expect you to make up the difference in random encounters as you trudge through the Underdark from point A to point B. On the plus side, there are many instances of quest XP awards throughout the adventure, although they tend to be small (which makes sense if they are merely a supplement to combat XP).
The chapters "don't contain nearly enough challenges" isn't a problem, since its not even an issue. I don't track any of this, and I don't have to.
You might be wondering how does it work, more precisely, and you'd be right to ask. The secret is: I'm not really handing out levels per chapters, or per monster/challenge/encounter/quest at all. I'm handing out a level after two, three or four play sessions (we play Sundays, so ours are probably longer than most. OTOH we probably play more meticulously and slowly than most)
As a DM, the only thing I need from the adventure is what OotA gives me. For instance, the adventurers should be level 8 when they escape the Underdark. Fine, I'll adapt the rate of levelling to match that.
Somewhat. Obviously they could make choices leading to an early departure, or a late departure. But if they stick around until they're level 9 that's not a big deal; it's just a matter of slowing down the level rate slightly. Perhaps have a level of four sessions (which otherwise is fairly uncommon).
One other thing: chapter 8 (or is it Ch 9?) is a "non combat chapter". Or, to be honest, it's an info dump

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Without checking my calendar, let's see... we've been playing for more than a year. Could that be 40 sessions if we play slightly more often than every other week on average? Looking it from the other side, I'm fairly sure we have spent no more than three sessions per level on average, and that's 42.
So, to put a number on it, I'm guessing 38 sessions to reach level 14

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While I can certainly see the old-school appeal of handing out actual XP, the illusion of objectivity is so thoroughly shattered for all of us, that's not enough of a reason to justify the added book-keeping.
(I can still remember how excited I was when I first got XP. Sigh, the joys of youth... It's just that now I'm so painfully aware its just a number. It doesn't mean anything, unless I reach the next level.)
I am continously toying with the idea of running an honest-to-gawd XP-for-Gold mini campaign, where you only get xp when you spend your illbegotten gains back in town (on carousing, building temples or whatever). (Saying this to show you guys I do see how XP can feel precious. It's just that I need an extra step you might not need. The way WFRP gives you an advance every 100 xp, would be another example)
"I think if I were running OotA I would make a more free-form XP system where every chapter contains 4-6 discoveries or accomplishments. Each time the players discover or accomplish 5 things, they gain a level."
To me this is just another example where you feel a need to distance yourself from the xp. "It's not me handing out xp, it's the book/monsters/table of accomplishments".
But one thing is for certain: Out of the Abyss is a singularly bad campaign to use if you absolutely must have xp. It's crystal clear the writers are using milestone levelling if even that. They're simply not interested at all in making sure challenges make up levels. I can't even imagine "they expect you to make up the difference in random encounters". My impression is that XP is simply an utter non-issue, and that they haven't even thought about it.
So for OotA: own up the fact that you the DM are the sole and arbitrary source of levels. Anything else and you'd be swimming against the current.
So, in a way, it's the ideal campaign for my way of awarding levels. They don't bother with bookkeeping - I don't bother with bookkeeping. A win-win. If you can stomach leaving xp out of it for a change

Zapp
PS. Then, it's a fact OotA lets you down somewhat in the latter half. Things get lots more sketchy, as if the writers are running out of steam, time, or both. Encounters fail miserably to keep up - except for Demon Lord encounters, you could probably stop leveling completely at level 11 and still breeze through all remaining monsters.
So to make ends meet, I've had to inject a couple of encounters of my own devising. Not that you don't do that as DMs yourselves. And this doesn't mean one way of handling (or not handling) XP works better or worse than the other. Just full disclosure: to carry the PCs through levels twelve and thirteen, I've used my own material.