Yes, the general rule that can be derived is that it's either-or: Either roll, or pick; not both roll and pick and then choose the best one or some other permutation that includes extra decision steps not described.

D&D 5e is written in "natural language", which is disparaged in some edition-warry parts of the Internet right now, but only means that the game is written so that the most obvious reading is generally going to be the right one, so that the game is understandable to the majority of readers who never read errata or participate in online debates about RAW, RAI, LFQW and other jargon.

In a natural language reading of all these rules, you either pick or roll, not both. If you were meant to do both, it would clearly say that, because not clearly saying that you're supposed to do both would be misunderstood by the majority of readers, if that was the intent. Since it doesn't say to pick and roll and take the best, the normal interpretations of "or" and "alternatively" are in force: either-or.

Extending the general principle of "when the game says to pick or roll, you pick or roll but not both" to other circumstances in the rules is a non-starter, because it just doesn't apply to any rules that it doesn't already apply to.

That principle isn't what your last question assumed would be derived though, so for completeness, let's now address the general rule that it assumes could be derived: that any roll could be instead replaced by taking the average, maybe if you want to save time or reduce randomness.

Extending "take the average" to other parts of the game where it's not mentioned as an option could be done as a house rule. It would probably not break most uses, and would likely find some corner cases where the extra predictability would break things and produce something that is off the average, but you'd have to play with the house rule for a while to find those, as is usual houseruling procedure.

• Off the top of my head, it would definitely break to-hit rolls. If nothing else, this is the clearest counter-evidence to suggest that it could have been meant as a derivable general rule.
• As an example of taking the average producing subtly skewed, non-average results, a wild magic sorcerer who knew that they could take the average damage of a surge effect might end up risking surges more often, thereby producing them more often than they would on average if all the involved rolls were actually rolled.
• Similarly, not rolling for healing means you can attempt to plan farther ahead knowing not only your current resources exactly, but also one step ahead into the future of your resources with reasonable accuracy. This would alter decision-making, skewing play in who-knows what different direction.

It's complicated

While skoormit has pointed out that the average percent increase for 1 die is 21.43, the overall increase for multiple dice is not straightforward, and highly depends on your Charisma modifier, more so for more dice.

I used a small python script to roll xd6. The lowest c dice (where c = positive Cha modifier) were re-rolled, unless there were less than c dice below average. The following data was gathered by averaging over 100,000 of such sets for each combination of x and c.

It shows the percentile increase of the average roll, i.e. ((new avg / old avg) − 1) × 100.

As you can see, the increase drops rapidly below 20% for x > c, though the slope becomes less steep as c increases.