That acceleration does not produce changes in time is well known. Time dilation of particles moving in circular particle accelerators can be precisely calculated by using only the velocities and completely ignoring the acceleration. This also is clear in the so called twin paradox. The time dilation of the traveling twin cannot be explained by acceleration. The acceleration can be made to be instantaneous then the only difference between the twins is the velocity.
Let us look at the twin paradox. In this modified version twin A (Traveling twin) and twin B (stay home twin) both initially start the journey together and undergo a large but brief acceleration to achieve the same large velocity. Twin A continues journey at a high velocity while twin B after achieving the same velocity immediately undergoes deceleration and turnaround acceleration and deceleration to stop at home. Twin A completes a high speed journey at a uniform velocity for a certain prolonged period then does exactly similar deceleration acceleration (as twin B) followed by return journey and deceleration.
By design the acceleration deceleration of both the twins is exactly the same and also very brief. The twins only differ in the time spent in traveling at uniform velocity. From here we can see that time dilation is only related to motion through space including the velocity present during acceleration. Also we know clearly from Lorentz’s and Einstein’s equations for time dilation that only velocity is involved and that time dilation is not caused by acceleration.
The same logic should be applied to acceleration in gravity. It should be clear from above that acceleration in gravity or in motion does not produce time dilation. Around large masses differential expansion of space creates time differential which then produces gravitation acceleration.
From our hypothesis and from observation we can see that all motion is due to expansion of space. Also if total motion imparted by expanding space is a constant then we expect to see slowing of time in an object if its external motion is increased. Similarly if a mass composed of billions of atoms moving at tremendous velocities is placed in a time a differential (gravity) it is natural to expect that it will move towards slower time and the internal motion will get converted to an external linear motion.
Twin paradox as well as the problem of acceleration seen in a rotating object is present only if you deny motion through space. Just as a moving object can be differentiated from a non moving object because of its slower time and so can a rotating mass be differentiated from a non rotating mass due to presence of acceleration. Objects set into motion produce a curvature in space while objects at rest do not. If Einstein had applied the concept of curved space to motion then there would not be any talk of twin paradox.