First of all, thank you all for replying and trying to help, I appreciate it.
So short update:
Today I took all the stuff I have been keeping in my spare room to fight reverb and put it in my living room as you can see in the attached photo (just for testing).
There's much less echo now and it improved the sound.
I feel like putting the rug closer to the speakers and getting the table out of the place were the most significant changes.
I think I can hear more bass now which makes me feel like the sound is more full.
Another interesting discovery - I use the TV as the source (or the amp's Bluetooth) and it turns out the volume of recordings in the Apple Music application on the TV is significantly lower than the iPhone's application. That's super weird. So now I just airplay from the iPhone to the TV or using the Bluetooth of the amp directly.
I also realized that prior to a few hours ago, I had a 'sound check' setting on the iPhone that might have messed with the tonality, IDK.
Although I recall that I couldn't tell the difference between streaming from iPhone or using the native app in the TV prior to today (their volume happened to be similar because of that volume limit setting that was enabled).
I do have something like 5 days to return the amp if I want.
Now I wonder if should just EQ the system to my liking (after acoustically treating the room) or replace the amp, maybe with a tube one or Arcam SA20.
My options are quite limited where I live.
The thing I liked about this amp is that its price is very competitive here. Some products from other places of the world can cost double the price for absolutely no reason.
So I will further test that amp but I am still wondering if I should just replace it or EQ.
You are right. In the soundbar, I have 'woofer setting' and it was all the way up (but frankly, I don't hear THAT much difference when it has a 'neutral' woofer setting).
And yes, I love the sound of the Airpods Pro.
A good start. One look at your room and the problem was obvious. Basically all reflective surfaces, very little absorption.
You came from a soundbar with probably not very much treble and lots of bass, to the KEF which has pretty good treble and even dispersion.
So in comparison it sounded like a knife in the ear.
I would say if you can get more heavy drapes, rugs, furniture, or even acoustic foam it might help a bit. However, if you go too far with this, you will quickly lose treble (and only treble) and the whole thing will sound boxy / muddy.
If you think the sound is OK right now, I would say to use some EQ to tweak the sound a bit more before adding more acoustic treatment. It is easy to overdo it with acoustic treatment for treble.
The most affordable way I know of to add EQ is this:
https://www.audiosciencereview.com/...lladsp-set-up-instructions-for-newbies.30830/ (costs around $50-100 US)
However, you can get a MiniDSP 2x4 for about double that.
Don't bother getting a different amp, it won't help unless it has tone controls... but tone controls are basically the same as using one of these EQ solutions.
A tube amp may or may not help... they are not all inherently warm, the effects are hard to predict, and the amps themselves tend to be expensive. I would try a tube amp last of all your options, even though they look cool.
You can also use a mini PC and use EQAPO, which is what I was doing recently. Pretty much any windows PC will work, and you can often find refurbished mini workstations for very cheap on Amazon. (example here is one under $90
https://www.amazon.com/HP-Prodesk-600-G2-Computer/dp/B082MPZ839/) EQ is very easy for a PC to do, so you don't need anything fancy. From there you need toslink IN and OUT, (and a DAC, probably) which can be a little harder to find, but the unit referenced in the CamillaDSP thread should work also.
This is all good for a 2.0 or 2.1 setup in most cases.