It is quite difficult to visualize and it took me a long time to "get it".
The path of the sun's annual motion relative to the stars is determined solely by my physical progress in orbit around the sun. A planet's axial tilt only changes the 'direction' I'm looking in and thus the reference point of my celestial coordinate system. The sun's path will always be a great circle on that celestial sphere (and always the same relative to the fixed background stars) regardless of which reference frame I choose. I think this is enough to surmise that the sun's angular speed on the celestial sphere is constant regardless of axial tilt (assuming a perfectly circular orbit).
Taken to an extreme, imagine a planet with 90° tilt -- the sun would move vertically and pass directly over the pole, making a constant 'horizontal' motion literally impossible.
I'm not sure what your tidally locked example is meant to demonstrate, since that's literally what the analemma is -- the path the sun would make in the sky once you subtract a planet's local axial rotation, i.e., tidally locking it to the orbital parent.
Thank you for the further explanation - this has convinced me that it's plausible, though my mind is slow and I still need to think about it to be fully convinced.
The path of the sun's annual motion relative to the stars is determined solely by my physical progress in orbit around the sun. A planet's axial tilt only changes the 'direction' I'm looking in and thus the reference point of my celestial coordinate system. The sun's path will always be a great circle on that celestial sphere (and always the same relative to the fixed background stars) regardless of which reference frame I choose. I think this is enough to surmise that the sun's angular speed on the celestial sphere is constant regardless of axial tilt (assuming a perfectly circular orbit).
Taken to an extreme, imagine a planet with 90° tilt -- the sun would move vertically and pass directly over the pole, making a constant 'horizontal' motion literally impossible.
I'm not sure what your tidally locked example is meant to demonstrate, since that's literally what the analemma is -- the path the sun would make in the sky once you subtract a planet's local axial rotation, i.e., tidally locking it to the orbital parent.