We added rugs to a couple rooms to improve the acoustics, we went with thick wool rugs to absorb some of the high frequencies, not just music. Swapped thin window treatments for thicker and more absorbent ones as a second step. Far from perfect but the house no longer echos every noise.
I’d suggest running TruePlay to see if it can improve your sound as a first step. Then try toggling it on and off to see if you prefer the adjusted sound over the default. If you find the sound still a bit bright with TruePlay on then adding some absorbent materials may help.
A rug isn’t your only option, wall and ceiling treatments are also available. Looking at your layout a larger/thicker rug might not be feet friendly and adding a thick absorbent under-pad would make that worse.
There are some good websites on both noise reduction and acoustically tuning rooms, some just tips, some selling products. I bought stuff from these (I think they are the same ones) folks many years back.
https://www.soundproofcow.com/