The trouble with viewing only log on the set and lighting for that range is that you'd be trying to make the image look "correct" for log in terms of the whites and blacks. In particular, since "white" on a grey scale falls around 65 IRE in something like ARRI Log-C, your tendency would be to try and get your whites up to around 80 or 90 IRE just to look better on a monitor.
Think of it like shooting film in the old days -- the negative has more dynamic range than the projection-contrast print made from it, but when you tested the film stocks before beginning production, you viewed prints, you didn't project the negative.
So in your tests, you might have found that in the print, you had about five stops over and five stops under before things burned out to white or fell off into black, so about 11-stops displayed for a negative that holds about 14 to 15 stops.
So this is very similar to a Rec.709 gamma applied to a log gamma signal. The Rec.709 version is the "print" and the log version is the "negative". You light for the print -- what can be displayed -- but you color-correct from the wider range of the negative, which gives you flexibility to bring things up or down.
The only caveat is that on set using a Rec.709 or similar LUT to convert the log signal, you don't generally have access (or time to use) tools like knee compression, luminance keys, and Power Windows which allow you to pull detail out of the extreme ends of log when necessary, such as a hot lampshade or curtain sheers or glare off of a sidewalk. Generally you just switch to viewing log to see if that stuff is getting recorded, then switch back to your Rec.709-ish LUT.
The trouble with lighting for the full range of log is that it can't be displayed that way later, not if you want normal blacks and whites on screen, so you may be more gutsy if lighting from looking at a log image, but as soon as you set your blacks to normal black in the final color-correction, you'll be shocked by the shadow detail that gets buried, and if you try to lift them up so that they can be seen, you end up with milky and noisy blacks on display. So it is better to think of the information at the extreme ends of log as just being there to make sure your shadows and highlights can roll off more gracefully into black or white rather than abruptly, i.e. more like film less like video.
When I used to shoot PanaLog on the Genesis, I sometimes would view log on the set -- with PanaLog, it's a very mild log curve, you just have to remember that white is around 70 IRE and black around 10 IRE, so it's not horrible to view, but you naturally tend to overexpose when viewing PanaLog only because of the dingy whites. But with ARRI Log-C, it is even worse, it's more like 65 IRE for white and 15 IRE for black, which is what is necessary to fit 14.5 stops of DR between 0 and 100.
C-Log in the Canon C300 is similar to PanaLog because it doesn't have the DR of an Alexa, so you can get away with just viewing C-Log on a set monitor if you had to.