In spectroscopy, a wide slit will lead to lower spectral resolution while too narrow of a slit will reduce incoming flux from the object. From what I read, it appears as though there is always a balance between spectral resolution and throughput. I'm looking for examples when an observer may choose one "extreme" over the other. ie. a very narrow slit or a wide slit (let's leave slitless spectra out of the convo for now unless relevant). What types of scenarios would motivate an observer towards one of the tradeoffs?
You seem to have all the ingredients apart from the variables of what size your detector pixels are (either physically or binned in software/hardware) and the angular extent of the object you are taking a spectrum of.
The basic trade-off, as you say, is between flux and spectral resolution, but there are limits to that trade off.
You should not reduce your slit width such that its projected size at detection is less than 2 pixels. This would be undersampling and you will not get the gain in spectral resolution you hoped for.
There is no point widening your slit in the hope of more flux if it already encloses most of the light from the object. In other words, if observing a point source, you don't usually open up the slit much beyond the "seeing" width. If you do, this can actually damage your data because the spectrum you get will start to vary depending on exactly where the star was positioned in the slit. Definitely bad news for accurate radial velocities.
The exception might be if trying spectrophotometry where you really want to get most of the flux and not be affected by seeing variability or guiding errors.
So your extremes would be: Use a narrow slit to achieve maximum (2-pixel) resolution on bright objects where radial velocity precision is important and the seeing is good enough so that your narrow slit still accumulates enough flux. Use a broad slit if the seeing is really bad but you don't care too much what your spectral resolution is. Or open the slit wide if you want to do flux-calibrated spectrophotometry and avoid flux variation due to seeing changes or guiding errors.