Before messing with all that check the transducers, input and output. If there are crap mics or crap speakers then there is no way it will sound good in the house. After that check the quality of the instruments on stage, if they are all cheap then well crap in = crap out.
Checking the processing has already been mentioned.
Is there anything wrong with the recordings?
I hear their live recording, and it sounds, how should I describe, not as open or transparent, but not technically "problematic". Their external studio recordings don't sound like this.
House speakers EAW FR153Z definitely could use upgrade, but it shouldn't affect the recording. Perhaps its lack of pattern control causes unwanted bleed into the microphones. I just hooked them up with RCF NXL44a to improve the pattern control of mid lows. It will definitely sound better, but should also help clean up the live mics pickup.
They feel perhaps the preamps on the GLD/AR not as high end as their studio preamps, or the onboarding EFX are not like the studio EFX. I hooked them up with SQ and DX stage box along with Waves interface, so there may be incremental improvements in each of these. They truly wanted to go to DLive or Yamaha QL5, but I talked them into trying the SQ/DX before taking the plunge for a $20k mixer.
They are using wired Beta 87a for lead vocalists, and I felt there is room for improvement there, and I sent them KSM9HS, KSM9, & KSM8 to try.
I know, this is a shot gun approach, without me being there to pinpoint the weak link of the system in person.