Only the very high end tuners have selectable IF bandwidths. Usually fixed not sliding, only wide and narrow. Kenwood has some. I have a KT-8300 that has wide and narrow. Even the narrow setting using ceramic filters, is not narrow enough for strong adj carriers. 600T had 3 BW settings.
As for drift, that depends on how well the LO is designed against temperature related drift. Assuming the LO has no feedback to correct for drift.
Wide IF filter bandwidths are meant for stereo reception. if you use mono, you can tighten up IF BW considerably as you do not need to pass through the sideband info used for stereo decoding.
Were there any receivers out there that had on-the-fly adjustable bandpass IF stages?
The Si4735 radio I built has selectable IF bandwidths, all done in the DSP filter algorithms. You can leave it figure it out in auto mode or you can program a register to set it, from 150KHz on up. It has a digital PLL for VCO, locked to a crystal, there is no drift unless the crystal drifts.
The Si4735 also has programmable or auto stereo blending to reduce stereo noise/hiss when receiving weaker signals. Sansui had this feature in some higher end analog tuners. They took the detected signal level and used that to control a LDR which was in series with a cap across the L/R signals. It was crude and the filter slope was fixed. In the SI4735 the blend amount/filter slope are adjustable, can be set manually or auto, determined by the S/N ratio and/or Rx sig level. very sophisticated by comparison.