How to tell if Acoustic Treatment is Needed?

I have a 12x22x8.5 ft room with the audio on the long wall - for various reasons - facing the listening chair 8.5' away from front of speakers. The soundstage is excellent with the center clean and tight. The ceiling slopes from 7.5' to 9' upwards from speaker wall to wall behind chair. I have no complaints; I think the sound is very good, although with 60 year-old ears and not a lot of experience with high-end audio systems I don't know whether MY budget system can be better.

I do know that there are furnishings in the room that people say do create problems, such as a large glass-fronted picture behind the chair, a coffee table in front of the chair. The speakers are older Mission 762s with front ports and thin cabinets (similar to Harbeth/Spendor BBC style) which sound quite nice. They sit with their back edge about 20" from wall.

My question is: How can one tell - or, what do you listen for - in order to determine if acoustic treatments would improve the sound? For example, I've tried moving the coffee table away from the chair but couldn't determine a change in sound.