Should I rip out the carpet?

Just as an experiment before you rip out the carpet, place something hard and removeable on the floor in front of the speakers, perhaps some plywood. The first reflections of sound waves to reach your ears come from the floor in front of the speakers. If this helps, then you could take the further step of taking the carpet out.

You might want to try putting a wall hanging over your screen when listening to music.
I found my 96 inch wide screen made the upper mid range less focused and stable. The midrange as a hole seemed to move around. I touched the screen and could feel it moving ever so slightly when playing 2 channel music. I found putting something over like a tapestry would stop the issue & it has much more focus & clarity. I also see it with movies when explosions happen but I also have a center speaker that seems to keep the effect in check.

As for your carpet I agree with Markphd. Go to Home Depot and purchase some 4'X 8' 1/8 sheets of plywood, luan or peg board with out the holes. Try them on your floor to see how it will sound. After the test take them back to Home depot and get your money back. I did it and left my carpet alone.

This is also how I tracked down the screen as being my problem. I also found I needed sound planks to help with first reflections.

I would not remove the carpet. Look elsewhere.
Sounds like your room is "overtreated".

True speculation without hearing it, but you seem extremely overdamped, which might be fine for home theatre but not for 2 channel. Maybe a much thinner carpet. Besides, if you've got shag, you've got no swag. Unless you're trying for some vintage look, I think it's time to retire the shag......regardless of what it's doing to the sound.