Recordings both video and audio (board or otherwise) are tough. You mix for the room, not for a tape, and video recorders have awful mics that distort easily and rob the lows. Most 'audience grade' mics and recorders are pretty much the same.
We do get recordings of our gigs, and I usually expect them to be light on bass and light on lows. I also as a conceit play very loud and in small rooms don't get a lot of support from the PA. That's fine with me. I sound great out front anyway (I know, I've gone FOH for sound check) and know that often times the house sound will use a tiny bit of reinforcement in the deep lows I tend to cut out then they'll 'bring the rest of the band up' around me. Of course, I've got a rig that's designed to be able to do that and when I'm outdoors or on large festival or theater shows, I bring my volume down and work with the house to a greater extent. The people in my band WANT me to be very loud on stage, the sound men pretty much get what I'm doing and I have good relationships with them. On gigs where I have a borrowed backline rig and low volume it usually impacts the performance because I'm not the only one unhappy with my lack of presence on stage, and soundmen don't like to put lows in their monitors besides the drum fill.
If you see me standing on the drum riser, you will know its because that's the only place I can hear myself effectively.