Good question, if a little late!
This was a big debate in the early 1930s, when high performance tuned circuits started becoming practical (either through triple-ganged tuning or superhet) and the "short wave" (now medium wave or AM) band started getting crowded.
People discovered that tuned circuits could be "too good" selecting the carrier and the sideband components closest to it, allowing only the LF part of the audio spectrum to be reproduced. The result was a bass-heavy, dull sound accused of being "drummy".
Some attempts to fix this included the "Stenode" which claimed to maintain tight tuning "so that a signal only 1 kHz away from the carrier could be eliminated" without impairing audio fidelity.
By 1934 this was debunked as a fraud; (sorry I can't find online references to that!)though some radio makers still called a treble boost circuit to flatten the response a "stenode" circuit.
By that stage (1934) "sideband theory" was widely accepted, and as it was easiest to detune a filter to set the correct filter bandwidth once and leave it there, the superhet "supersonic heterodyne" radio took over from the simpler "straight" sets.
Naturally you had to compromise between high fidelity (this was before AM transmitters were tightly regulated as to how much spectrum they could broadcast!) and rejecting interference, and no one compromise would work for all situations.
So here is a serious hi-fi AM tuner, from 1947, the Leak VS where you can adjust the bandwidth of the IF filter to get the best sound from any station you tune. I heard some lovely Asian traditional music on the 19 metre band (about 15MHz) - turned out to be Radio Hanoi!