it is the speaker. it has a hollow presenting upper bass, lower midrange and couple that with the overly present lower treble. Makes for a bright sounding speaker.
You'd find this out if you invest in a measurement mic and use REW for sanity RTA measurements @lendivf
Also the vertical dispersion is narrow enough with flaring in the lower treble presentation with things sounding brittle and papery.
So answering you on what to tackle:
1/ room acoustic - Yes this is what to focus all your energies on
2/ change my class D amplifier for tubes or transistors - Nope, not the culprit
3/ find better cables - even less of a culprit
4/ me turning old ? - can be a factor but also low on the importance ladder.
Take the time to get the measurement mic. ge this one - miniDSP, UMIK-2 USB Reference Measurement Microphone
and get REW - Room EQ Wizard Room Acoustics Software
Use this video as guidline to set it up - Room EQ Wizard: Start to Finish Tutorial - Ep3
and follow this method to get the data needed to show you how your speakers are coupling with the room - Moving Mic Measurement | diyAudio