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You’ll feel the tides, however.

These analogies always fall apart for me because you'll be burnt to nothing long before you reach the surface of the sun.

Your analogy falls apart, too. Sun does not have a surface to be reached. It just sort of gets more and more dense and that's all.

What is "feeling"? If my senses can register changing spectra of the stars around me then yes, I will "feel" the acceleration.

I didn't think you could accelerate at all in a freefall (which is why you wouldn't feel it), you are just following a geodesic