Replying to your first paragraph. You're casually ignoring how it got to eigenstate a in the first place. The particle is in an eigenstate a because it was act

Replying to your first paragraph. You're casually ignoring how it got to eigenstate a in the first place. The particle is in an eigenstate a because it was acted on by the A operator. If it was in an eigenstate b prior to this, or any other state that isn't a single eigenstate of A, the action of operator A disturbed that original state. The HUP itself is a direct consequence of measuring the uncertainty of two noncommuting operators, and its derivation is a fairly straightforward, though a bit annoying, mathematical result.