"Does anyone else think the games would be better off without these?"
I don't play videogames so they can remind me about the real world. And I play Mass Effect in particular so I can feel a sense of accomplishment when I do manage to reconcile both sides of a conflict. I don't think of them as "easy way out" options, I think of them as taking the difficult path and succeeding, which is where Paragon playthroughs shine.
So, no, I don't believe the games would be better off without them. I believe it would just make them more cynical. If I want cynical, I'll play The Witcher.
The only time I feel it's a good idea not to have a "win-win" is in a case like curing the genophage, where to do the right thing your only sacrifice is the support of a party who believes holding military support hostage in exchange for screwing an entire species over is a good idea. This is actually one of the better moral choices in the series, since you do lose out on something practical by choosing to do the honorable thing, but the consequences of doing the honorable thing vary based on past choices in both ME1 and ME2 (if Wrex and Eve both end up dead you can even convince Mordin that curing the genophage is a bad idea, because frankly, at that point, IT IS). In this case not having a win-win makes sense, and still affords you an ending that allows you both the cynicism of a pragmatic betrayal and the warm feelings of winning as a paragon.
The ending itself, for all of its problems, does lack a "win-win" option, though. In the two endings that preserve all races of the galaxy, Shepard must make a morally dangerous choice and sacrifice him/herself. In the ending where Shepard can potentially survive and not make a great sweeping galactic change without the galaxy's input or consent, EDI and the geth are wiped out (this is confirmed in the ending--EDI's name ends up on the memorial plaque and all geth-related happy ending images are prevented from appearing). In other words, there's no perfect answer. And I am okay with that in this instance, since I can at least prioritize whose happy ending I want to give up--Shepard's or that of the synthetics in the galaxy. The Catalyst's choice is a part of the ending I actually have a relatively small problem with compared to everything else that happens in the last legs of the game...
" Why can't I leave her in the cage for the council to decide her fate? Even worse, later in the game, Kaiden chastises me for not taking this option, which the writer just didn't make available to me for no obvious reason."
Because they would have killed her, so it would be pointless; and because Shepard had no way of securing the lab to keep the Queen contained, so it would have been impractical. The Council gets on your ass about the Rachni decision no matter what choice you pick, but they never complain "You should have let us decide!" It's only ever, "BAW, YOU TOOK A BIG RISK" or "BAW, GENOCIDE!" You can't please the Council on that one. Which is why I usually leave Noveria for last... so it skips that Council call altogether.