Don't know what "high output" models you're thinking of mrvco, could you specify? For higher efficiency speakers marketed for home use (that often have limited power handling capacity) and single driver designs (best suited for small ensemble genres at limited sound pressure - volume levels) most listening is done at very lower power levels. Keep in mind that wattage versus dB is logarithmic, so it takes 10 times the power to sound twice as loud (a 10 dB increase). Such genres are normally enjoyed at say 85 dB for "spirited" listening. If the speaker is rated at 95 dB/w/m (another issue with vendor honesty) you'd be using 0.1 watts per channel is a small room and maybe 0.5 watts per channel in a large room.
On the other hand music is made up of peaks and it's those peaks that demand lots of power, due again to that logarithmic relationship. Classical music can exhibit 30 dB peaks, jazz 20 dB, and rock 10 dB. Live performances can peak at 105 dB for classical or jazz (more steady) and rock at 110 dB (nearly constant). So that same 95 dB/w/m speaker could need 100 watts per channel or more to reach live classical/jazz peaks and 325 watts to reach live rock levels. Thankfully speakers handle peaks rather easily (as long as the signal isn't distorted).
Traditionally solid state amps don't handle peaks as "cleanly" as tubes that ordinarily have big/heavy power supplies, but nowadays are easier to find at higher power ratings. On the flip side tube amps excel at lower levels where we typically listen.
Speaker impedance varies with frequency. The more benign designs vary less, but all rise steeply as it's lower frequency limit is reached and also commonly gradually rise at treble frequencies. If you look at several speaker impedance graphs you'll discover that impedance ratings are painted with very wide brush strokes (highly generalized outside midrange frequencies).
Typically amps handle higher impedances better than lower (especially tube amps). The mark of a "good" amp (stable, beefy, not fussy design) is one rated at double the wattage at 4 ohms versus 8 ohms. Lower impedance loadings can cause "weaker" amps to become unstable and be prone to overheating. In fact many tube amps have 8 and 16 ohm binding posts. If the amp doesn't mention 4 ohm output it's probably because it's not recommended. Note that historically Decware amps have been the exception, they thrive under low impedance loads.
Impedance versus wattage is a simple electrical phenomenon. If the amp is stable at 4 ohms, it's output will be doubled at 4 ohms versus at 8 ohms. Sound quality shouldn't vary for a given amp regardless of the load impedance.