Here's an abstract view regarding the inevitability of gimbal lock: The state of a single gimbal is described by an angle. Since angles are mod 360°, that is topologically a circle. The state of three gimbals are then given by three angles. One point on each of three circles is a point on a three-dimensional torus T^3. With no gimbal lock, you get a map from T^3 to SO(3), which is locally a diffeomorphism at every point. For topological reasons, no such map exists: Due to compactness, it would be a cover, but the only covers of SO(3) are SO(3) itself (a single cover) and the three-sphere (unit quaternions, a double cover). And T^3 is distinct from either of these two. Hence gimbal lock is unavoidable with three gimbals. (Four gimbals is a different story, but then you have a redundant dimension to play with.)
lol i'm a dummy for never realizing that SO(3) isn't homeomorphic (the diffeo part isn't necessary to prove this...) to T^3 (which i naively thought because s in SO(3) seemingly has 3 free parameters). surprise surprise SO(3) is actually homeomorphic to P^3 lol.
You can drop “seemingly”: SO(3) is indeed tree-dimensional. And SO(3) a.k.a. P^3 has fundamental group Z_2 a.k.a. GF(2), whereas T^3 has fundamental group Z^3, so they are quite different beasts indeed.
