I know others who share your perspective. I don't consider it the best stat for measuring overall excellence, but rather a quick-and-dirty way of considering whether the guy has done his job. In general, good pitchers are going to have a higher percentage of QS than bad pitchers. It's a bit like save percentage in that regard. It's kinda helpful, but doesn't give a full picture of how good a reliever has been. Or even wins. We can point to all kinds of reasons why wins isn't the best indicator of pitcher success, but in general better pitchers are going to have more wins that bad pitchers.
So while not a sole indicator of any kind, I think there is some value of having such a quick-and-dirty measure of effectiveness. In that you'd accept the modification of it being two runs in six innings, it seems you agree at some level.
So if it has some level of helpfulness, the question is what to use as the standard. If I'm remembering the original premise, there were a couple of key questions Bill James was trying to get at. The first is the definition of "quality." I highlighted your words "standard of excellence." I don't think he was intended it as something to count "excellent" stats. That's where the OP started to go with the notion of a "strong" start.
Rather, I think he intended to consider "quality" as "kept the team in the game, gave the team a chance to win, both that day and in the future." And if you think about it, a starter that gives up 3 runs in 6 innings has generally done that. Hopefully the team's offense can score that many, and he generally hasn't overextended the bullpen in a way that affects future games.
The second part is that as a "quick-and-dirty" stat is has to be easy. To be, it has to use round numbers rather than partial innings. So then, the reasonable permutations to consider are probably 3 runs in 7 (3.86 ERA), 3 in 6 (4.5), 3 in 5 (5.4), 2 in 6 (3.0) or 2 in 5 (3.6).
Mathematically, 3 in 7 or 2 in 5 may be the best of the bunch as minimums. In practical terms, however, few pitchers go 7. Almost by definition, going a full 7 in today's game, even if you give up four runs, is some level of excellence, just for the fact of saving the bullpen. Conversely, only going 5 is going to overextend the bullpen if it happens regularly, even if you never give up more than two runs. So 6 innings feels like the standard to use in terms of length.
If that's the case, the next question is whether to use 2 runs or 3 allowed as the standard, and here it goes back to the definition -- are we measuring "excellence" or are we measuring "kept the team in the game and didn't shoot the bullpen." If it's the latter, I think 3 is okay.
A final comment that I have found helpful from others is to think of what a season would look like if every start was a quality start. Not "every start was the minimum for a quality start" (i.e., each start was 3 in 6), but rather a season where the worst start was a 3 in 6. That starter would almost certainly end the season with good overall numbers, because they would have had some starts where they went 6+, even 7 or maybe an occasional 8. They also would have had starts with just two runs, even some ones and maybe an occasional 0 runs. Each of those would push the overall ERA down. I'm sure it's out there somewhere, but I wonder what the ERA is for pitchers if you only count their "quality starts." I wouldn't be surprised it's 3.50 or better.