Whether or not you're right about your anecdotal observation, there are some reasons not to make the feedback circuit tweakable with a variable resistor. At least one some "front panel" one intended as a primary control.
One reason is the stability considerations of the amplifier, which vary with the amount of feedback. The greater the feedback, the more difficult it is for amplifiers to be stable. That is why you hear amplifiers being touted as "unity gain stable": it means that even if you configure them with a gain of only 1 (as opposed to 2, 3, ...) they will still be stable. If the feedback circuit of op-amp that is not unity-gain stable is allowed to be controlled by a variable resistor, then it could be taken into a region of unstable operation.
Secondly, see this question about noise gain. The noise gain of an op-amp follows the same formula as the non-inverting gain, whether or not the topology is inverting or non-inverting. If you try to use the inverting configuration with a variable resistor such that you allow gains below one, you actually end up with a situation in which you have signal gain less than one, and noise gain greater than one. That is not to say that a voltage divider after the output is noise free. Sure the noise relative to the signal rises as it as attenuated toward zero, but at least that passive network isn't actively amplifying noise while attenuating signal.
These aren't reasons to reject the idea of having some variable control in the feedback of amplifiers. In audio amplifiers, it has been a common technique for decades to implement tone controls in the feedback circuit. See the 1952 Negative Feedback Tone Control paper by P. J. Baxandall for instance.
