In my opinion: Euler did so much already that the contributions from later physicists/mathematicians are at a level of abstraction that is beyond the scope of standard physics courses.
Among the subjects not already covered by Euler, I assume, is the quite unique case of the intermediate axis theorem. Arguably awareness of that phenomenon took off only because of the video recorded on a space station showing a demonstration of the motion of a rigid body with three different moments of inertia
The equations for that case are known, numerical simulations reproduce the motion. But will a textbook author include a section about the intermediate axis theorem? I doubt it; even the minimum mathematics to treat the case is already very abstract.
(Also, in order to move according to the idealized model the object must be perfectly rigid. But in the real world there is no such thing as perfect rigidity. Any mechanism that can dissipate kinetic energy will dissipate kinetic energy.)
So, in the end: the part of rotational mechanics that makes it into physics textbooks has a short history: it's Euler.
As to development of rotational mechanics prior to Euler:
Newton had inferred that since the Earth is rotating it will have an equatorial bulge. Because of that non-spherical shape the center of gravitational attraction does not coincide with the center of mass. (The two coincide only in the case of a perfectly spherical celestial body.) As a consequence the gravity of the Sun exerts a torque on the Earth, and so does the gravity of the Moon. Newton had arrived at the view that the precession of the equinoxes was due to this combined torque effect from Sun and Moon.
The interpretation must be that Newton had anticipated the concept of gyroscopic precession. Not a generalized concept of gyroscopic precession, but a concept specific for the Earth precession. In the Principia Newton offers a calculation.
The outcome of Newton's calculation matches the actual precession of the equinoxes quite well, but historians of science state that the accuracy of the data that Newton was working with was certainly not good enough to be that succesful. There is a question here on HSM about that Was Newton's succesful calculation of precession of equinoxes a fluke?
Most of the Principia was later restated in the form of differential calculus. It was that more accessible form that spread throughout the physics community. It would appear: on the subject of the precession of the equinoxes Newton's description in the Principia was not enough to reconstruct his thought process. So it appears that on that subject there was no follow-up.