Well, the second paragraph really doesn't describe this series at all. You say that the availability of Kopech and Crochet is forcing guys into multi-inning roles - but take a look at the voyage this series and how it turned into losses and that doesn't show up. Rather, it all hangs on LaRussa.
Bummer pitched today on normal rest for a starter - he last pitched last Tuesday. Heuer pitched today on the same rest. Hendricks pitched today on 3 days rest - he threw an inning on Wednesday. Now really, with that context, was it the unavailability of relievers forcing guys into these roles? At least on Friday, they were literally all available, but LaRussa was saving Bummer, Hendricks, and Heuer for what, the 13th inning? Marshall for 2 innings was totally LaRussa's choice.
Furthermore, and this one's rather big - LaRussa went to Foster on Saturday with the Yankees up 3 and Foster got 3 outs in 7 batters and the lead doubled. Fine, maybe the White Sox weren't hitting Gerrit Cole, and it didn't matter whether the lead was 1 run, 3 runs, or 6 runs, so bringing in Foster and waving the white flag wasn't worth getting bent out of shape about. But then who else did they use once the game was out of reach, in a 7 run game in the 8th inning? Crochet. So, what exactly forced the White Sox into using all their relievers today and then having to plan for how to have a pitcher available in the 10th? Part Keuchel being blah, but part the fact that LaRussa wasted Crochet in the bottom of the 8th inning of a 7 run game the day before, after he had already waived the white flag in the game by pitching Foster, so they couldn't call Crochet out for 2 innings on Sunday if they had needed him.
So is it the restrictions on Kopech and Crochet that are causing odd usages, or is it LaRussa himself? At least this series, LaRussa.
One other point. In 2019, Liam Hendriks pitched 85 innings As of right now, Hendriks is on pace to pitch 65 innings. The only one on pace for more than 68 innings out of the White Sox's bullpen is Kopech. So, could he, one of their better relievers still, be available for more? Frankly, darn right, and that could very well make this bullpen better.