As an aside, many AVR's also have "tone controls" - on their remote, or on their Smartphone app control - what they tend not to have, is the physical knobs on the front panel.
eg: The current Integra/Pioneer/Onkyo's have treble, bass, and "vocal" buttons on the remote.
As others have said - the biggest impact on your sound will be your speakers and the room (and the speakers placement IN the room)...
Once you have chosen your speakers, you can then choose your Integrated... (a receiver IS an Integrated amp! - just more stuff is integrated!
)
In my case, the current AVR I chose did not have sufficient power/current to properly drive my speakers, so I paired it with external power amps.... the AVR it replaced was higher powered and handed the speakers fine. (hence I chose an AVR with pre-outs so I could run it as a sophisticated Pre-amp, with my power amps)
The other thing that many AVR's bring to the table that typical integrated's don't - is proper Room & Speaker DSP correction.
Don't underestimate the value of good room/speaker EQ it is a paradigm shift, and substantially improves the results achievable in our typically imperfect living arrangements!
Good AVR's have good DAC's, good preamps, and a wealth of tuning options - they are the complete opposite of minimalist analogue preamps/integrateds.
But if you want to move into today's world of bluetooth, wifi, HDMI etc... then minimalist analogue is not going to get you there!