The Eldar aren't designed as bio weapons. The Eldar are designed as vassals. They are designed as vassals that let the Old Ones dick around with the psychic powers tech trees on someone beside themselves. The Eldar are vassals that send out Janissary to fight for their lords and masters.
The major issue for the Old Ones is that they don't diplomacy. The Old Ones failed so hard at diplomacy the died of it (turns our that 1000 years at 100%+ war exhaustion for a race with life spans measures in 20-30 years Max means they go crazy nuts, who knew?), the Eldar had no one riding herd on them. They are genetically designed to require someone riding herd on them. Left to their own devices they party until the universe breaks. Literally.
Basically, the Orks never had a not-war mode installed or an off button installed (also not actually finished). The Eldar never were designed to be in charge of themselves. Avernus is a never ending adaptive weapons development lab. The Old Ones second biggest failing was never designing anything that didn't require constant attention to work properly.
Its a personal theory of mine the tyranids are also their work. Bio-weapon that does one thing forever to the point of stupid and ruins everything for everyone around them. That he Squat apparently got eaten by a hive fleet coming from the center of the galaxy and that Fenris had some support this theory.
Actually the Old Ones didn't diplo fail, at least according to what we know now.
Apparently the reason the Necrontyr went to war wasn't because they were dying so quickly (well that was the stated reason), but because the Triarch realised that they needed an enemy to keep their people together. As the strongest, but least likely to kill them all species was the Old Ones they went for em (if you play Stellaris it'd be like attacking the Enigmatic Observer Fallen Empire at mid game, or to translate a nice way to waste a lot of ships and time, but there's certainly worse precursors to pick a fight with.)
Now of course the Old Ones did stick em on their world, but look at it from their (presumably alien) perspective. The Old Ones are ancient beyond belief, presumably they saw this and thought "another tantrum by a younger race...well lets discipline them, let them get their perspectives back in order and then let them get back to doing the space thing, lets be the enemy for them to unite behind...they'll grow out of it" because I refuse to believe a race intelligent enough to create Necrodermis before they even got off their tomb world of a home world couldn't increase their lifespan as well.
As for the Eldar's exact relationship with the Old Ones, well that's still complex as they don't quite know themselves. I do agree with you that they likely weren't intended to act without the Old One's guiding hands for at least a few more centuries, and I even agree they were an experiment (specifically in making created Gods, what with all the weirdness going on with their pantheon.) However, their mythology says they weren't used as a warrior race until the second half of the War in Heaven (basically when they were loosing). However, the info we have implies that the Krork and Joekareo (races made specifically for war) weren't created until after this, so I think my original assertion still stands that as the Old Ones got more and more desperate they started shedding more and more morals for survival. Thus the Eldar being more...well rounded than the Krork, but also more of a rush job.
Actually I think you're underestimating the Old One's abilities on the whole bio weapon maintenance thing. The Eldar managed to last at a minimum 60 million years before finally blowing up (and despite that have managed to survive...to an extent), the Orks do actually have an off button they're called the Krork...thing is nobody expected
all the Krork to end up dead? Also they were finished...in embers the Krork even have the mandate of save life in the galaxy...albeit by any means necessary...just again the Orks degenerated when the Krork were wiped out (canon regardless of embers or not). As for Avernus, well yes it does require constant attention, but they made sure that either it has an overseer that is not only adaptive, immortal and regulating, or made the planet itself all of those things (depending on how the Planet Mind works). Not bad for people who've been died out around the same time the owl was only just starting to come into existence.
As for the Nids...eh I can see the arguments, but I find it unlikely. First even though they'v both forgotten a lot the Nids seem like a pretty big thing to forget unless the Old Ones made them in another galaxy entirely...which then posses the question when did that happen? Regardless the Nids are implied to have been firing shit at other galaxies for ages, turns out they even landed an entire hive fleet in the galaxy in
M35. Seeing how they were introduced as spores I think things like the Catachan devil and Fenrisian Kraken being introduced the same way as Tiamat and the genestealers makes sense.
Also on the Squats, Geedubs has reversed the ruling...well they did that a while ago actually. They first killed the Squats off with a hive fleet, but then they got rid of that and pretended they never existed, though they were confirmed as being a thing in 6th edition codex. They're back officially now in some capacity as Geedubs is now selling models again.
