I know about this suggestion. Mine is an even further extension of that, and I'm asking about whether there are any further issues and whether I've fixed Lucky.
I find the way 5e handles multiple forms of Advantage disappointing.
It has prioritised "simplicity" so far that it removes a lot of strategic options from combat—specifically the fact that a second source of advantage grants no benefit, and having lots of advantage is entirely negated by a single disadvantage (and v.v.).
Would this variant rule cause problems?
Variant-Multiple sources of Advantage or Disadvantage.
For each source of Advantage or Disadvantage, roll an additional d20. Then, for each source of Advantage discard the lowest current die, and for each source of Disadvantage discard the next highest die.
Variant-Lucky Feat with multiple sources
If a character with the Lucky Feat rolls with any form of disadvantage (and chooses to use their Luck point), then they roll an additional die and discard any single die of their choice.
This reduces down to standard Advantage or Disadvantage, if there's only one source of (Dis)Advantage.
It uses this version of Lucky+(Dis)Advantage from the Sage Advice Compendium:
If a DM wants advantage and disadvantage to play their normal roles even when the Lucky feat is used, here’s a way to do so: roll two d20s for advantage/disadvantage, roll a third d20 for Lucky, eliminate one of the three dice, and then use the higher (for advantage) or lower (for disadvantage) of the two dice that remain.
And it reduces down to this suggestion, for the case of one Advantage and one Disadvantage.
I imagine that the conclusion is likely to be "same as the answers to this suggestion, only more so."
But I want to check there's nothing I've overlooked and that I've fixed the problem with Luck.
Examples:
3 Advantage (Roll 4 dice, take the highest):
4, 6, 7, 18 would become 18.
3 Advantage 3 Disadvantange (Roll 7 dice, take the middle):
3, 5, 6, 10, 14, 20, 20 would become 10.
2 Advantage 4 Disadvantage (Roll 7 dice, take the third lowest / fifth highest):
1, 4, 8, 13, 17, 19, 20 would become 8.
Best Answer
I tend to agree with the other answers and comments saying that allowing advantage and/or disadvantage to stack risks slowing down the gameplay.
Such mechanics can work in some systems and contexts — there are entire rules-light RPG systems whose core task resolution mechanic boils down to "list every narrative circumstance that could give your character an advantage here, then roll that many dice and take highest" — but the advantage mechanic in D&D 5e is deliberately designed not to allow this kind of circumstantial bonus stacking. Before breaking such a deliberate design decision, I'd recommend discussing it with the entire group you're playing with and making sure that eveybody's OK with it, and I'd also consider playtesting it in a quick one-shot game or two and having another discussion afterwards before adopting it for a longer campaign.
That out of the way, let's look at the mechanical effects of your suggested rules. To help with that, here's a simple AnyDice function to model your mechanic:
All it does is simulate rolling \$A+D+1\$ dice, skipping the highest \$D\$ and taking the next highest. It doesn't explicitly simulate your luck mechanic, but that's in most cases equivalent to simply adding one source of advantage. (It's slightly more situationally flexible, since a player could in principle choose to discard some die other than the lowest. But if they want to roll high, as they usually do, then they won't.)
Anyway, here's what jumps out at me:
Yes, your modification to the Lucky feat fixes the problem noted in this answer to the earlier question. In fact, it makes the feat strictly weaker than in vanilla 5e by eliminating the ability to spend a luck point to turn disadvantage into super-advantage. I cannot comment on whether nerfing the feat like that would be a good or a bad thing, except to note that if any of your players currently have the feat, you definitely should discuss it with them.
As you note, your mechanic matches vanilla 5e behavior in the common cases where there's at most one level of advantage or disadvantage, and not both at the same time. Thus, if players did not actively seek out advantage (or disadvantage for their opponents), your homebrew rule would likely make a difference relatively rarely. But of course they do, and will likely do so even more enthusiastically under your proposed rule.
In terms of the average result, stacking multiple levels of advantage (or disadvantage) gives diminishing returns. But the effects are far more pronounced at the high (for stacked advantage, or low for disadvantage) end of the d20 range. In particular, for the first 5 or so levels of advantage, the probability of rolling a natural 20 is almost directly proportional to the number of advantage levels you have, and only diminishes slowly with further levels. This makes stacked advantage super valuable for characters aiming for crits.
When rolling with both advantage and disadvantage, your mechanic results in a distribution that is peaked at the middle (biased up or down depending on which side dominates) with a very low chance of either very low or very high rolls. In the limit with \$N \to \infty\$, with \$N\$ levels of both advantage and disadvantage, it effectively tends to a "take 10" distribution concentrated at 10 and 11, with a very small chance of rolling much above or below those numbers.
(In particular, as noted in Thomas Markov's answer, both natural 1 and natural 20 are very unlikely results when rolling with both advantage and disadvantage, unless you have a lot more of one than the other.)
Regarding the last point above, traditionally in D&D such low-variance mechanics have been reserved for low-risk, low-pressure situations where characters are allowed to take their time with the task at hand. That seems like the exact opposite of a tense combat or challenge situation with multiple counteracting influences imposing both advantages and disadvantages. It thus seems counterintuitive to me that stacking a combination of advantage and disadvantage should make the outcome of the roll more predictable, as happens under your proposed rule.
If anything, if I wanted to implement stackable (dis)advantage in my game, I'd therefore prefer a rule where simultaneous sources of advantage and disadvantage would cancel out (much like they do also in vanilla 5e), so that e.g. three sources of advantage and one source of disadvantage would equal \$3-1 = 2\$ levels of advantage. This would also simplify your house rule slightly (since you'd always take either the highest or the lowest roll) and reduce the average number of dice rolled.
Ps. As a slight tangent that I can't resist including, one could even consider rules where simultaneous advantage and disadvantage would make the roll more swingy, favoring very low or very high rolls over the middle. One way to implement such a mechanic, at the cost of some further complexity, would be as follows:
Perhaps surprisingly, this complex and strange-sounding mechanic produces very nice distributions of results and achieves its goal of favoring the ends of the d20 range when rolling with simultaneous advantage and disadvantage. Whether it would actually be fun to use in a game would, of course, be another matter entirely.