Clarification on circular motion and pseudo-forces

You are correct that pseudo-forces can be viewed as a mathematical correction to allow for Newton's second law to work in non-inertial reference frames, and I tend to think of it as such most of the time. However, really these pseudoforces are the effects of inertia. They aren't forces, but they still are a property of matter.

I would argue then that you really do "feel" pseudoforces due to your own inertia. You have already given the example of circular motion. Another example is when you go upward in an elevator you feel heavier for a little bit, and then when the elevator slows down you feel lighter for a little bit. Sure these effects are not forces (as they do not follow Newton's third law), but they still are "real" in terms of they arise from something physical.

I don't think there is any law saying forces are the only thing you can "feel". Especially if your body is actually in a non-inertial frame of reference.