I think the modern definition of "onsight" is a silly adaptation of the rules for climbing competitions.
The old-fashioned idea ("trad onsight?"), was that whatever knowledge you acquired was obtained from the ground up and without weighting any gear. You could climb up, look around, downclimb and reclimb to your heart's content as long as you started at the bottom (no traversing in from the side or climbing down from the top to inspect) and didn't weight any gear.
Downclimbing out of trouble before falling is one of the great (and perhaps now fading) trad arts. The current sport-influenced definition of onsight misses the boat and encourages rushing upwards before you blow it.
I'm not sure what you think onsight now means? Your definition of what it "used to" mean is exactly what everyone I know considers it to mean. Unless I'm missing something.
What's the "sport influenced" version? By this do you mean that the climber always continues to either move up or fall? That's silly - that's just an onsight attempt with poor tactical skills. Milking rests and up and down-climbing to suss out a crux is just part of climbing, neither old- nor new-school.
Regarding the original topic - at the Gunks I preferred to attempt to onsight every climb that was challenging to me. This wasn't difficult logistically, since I consistently climbed with either people much weaker than I - who could at best follow the stuff I could lead, or people much stronger than I - who's leads I could at best follow.
These days, sadly, my only trad partners are weaker than I, and mostly less knowledgeable. So every hard climb I get on is an onsight attempt.
GO