If something's too loud in the house, it's going to be too loud on the
recording, so you might as well turn it down. If it can't be heard in
the house, it'll requre a lot of boost when mixing the recording so
you might as well turn it up.

Yep, that's the facts. I used to work live sound. In most clubs with
an electric band there were often faders that weren't even in the mix
or were down 20 dB. Such as bass or guitars. In that case the noise
floor was a mess on the recorder end out of the direct. The only way
it ever worked for me was to use the inserts and patch the recoder
into the insert chain before any effects. A wiring PITA but doable.
However if you're plugging into someone else's rig then you'll have to
play by their rules or hope they're both willing and able to hook you
up. Another trick is to use subgroups if they're available. Set the
input channel to unity gain and use the channel direct out to the
recorder, buss it to a sub and use the sub fader for the house mix.