What algorithm will solve IK?

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You say the IK solver doesn’t interact well with constraints, it doesn’t at all in reality is my experience. You also say it doesn’t interact well with drivers, that I dispute as I have used many drivers in IK constraints and they work well provided you don’t try to override the limits set in the IK chain.

I modified my rig to replace the constraints along the IK chain with drivers and it seems to be working. I set up my rig so that both shoulders have three bones: ShoulderDriver, ShoulderManual, and just Shoulder. I have a constraint on Shoulder to add the rotation of ShoulderDriver and ShoulderManual. UpArm is a child of Shoulder, and Shoulder has a driver that depends on UpArm. I think this would be a dependency cycle but Blender is not warning me about a dependency cycle. Are dependency cycles involving drivers not an issue?