This does not look like a terrible room to me. Ideally, nothing should be within the triangle formed by the speakers and listening position, but this is a lovely, comfortable family room, so compromises are sometimes required. At least the coffee table is not made of glass!
By my math, the mic position was 3 7/8 inches closer to the right speaker. That, or REW just had a little trouble working out the timing offset. Did you use an Acoustic timing reference when you made your measurements? If not, give that a try for your next measurement. Also, when you include the REL, do separate plots of Left + REL and Right + REL. Including all three can be interesting to look for phase issues, but it's more useful to have plots with each channel separate when considering possibilities for corrections.
A number of plots have "Herbie" in their description, but I could not figure out what that means. Also, the soundcard calibration for those plots looks really crazy, so I decided to ignore them for now.
Looking at the first few ms of the step function, I see a second hump of nearly equal amplitude to the original impulse but delayed by 1.39 ms. This indicates a strong reflection with a path length that is about 19 inches longer than the direct sound. If you can figure out what that is, removing it or covering it with something that will scatter or absorb will improve things quite a bit. I'm hoping this is not the aforementioned coffee table.
I'm not aware of any digital room correction process that will remove a reflection like this, so like others have suggested, it has to be dealt with using practical means.
Overall, the RT60 plots look pretty good. Rough average is a little over 400 ms for most of the range. It's common to lose articulation below 100 Hz in a room like this with no bass traps. A pair of
these would do wonders here and be well worth the investment if they won't spoil your decore.
The frequency response of the left and right channels looks good apart from the broad ~8 dB dip from 150 to 500 Hz. Tonality will improve quite a bit if this can be smoothed out using EQ.
I am a little concerned by the ~2.5 dB channel imbalance between 2 and 5.5 kHz. That covers a fairly wide range where our hearing is sensitive to stuff like this, so it may be worth increasing the gain in the left channel by a couple of dB to see if that helps.
I didn't study the plots with the REL included, but it looks like you should be able to bring it in at around 60 Hz, give or take, to add extension and pressurize the room.
I see that you have a turntable, so I assume that digital room correction is off the table (no pun intended). Honestly, I don't think it's required to make this room really sing, but for nice tonality, we really don't want to be down at 250 Hz. Hoping you can find a way to address this that works with your room and the rest of your system. Hope this helps. Good luck!