Micing will give the character of the rig, but the DI will better give the attack and articulation.
I think the booming muddy mess problem is that when they turn the bass up in the PA it adds to the already too much boom from the bass rig, and many don't seem to know that they can use those little faucets on the board to remove the boomy frequencies from the bass channel and to add growl and/ or bite to the bass channel. But, if there is already too much of something in the room from the bass rig, they cannot cure the problem, only try to fill in the missing bits and hope the end result is not too loud for the mix.
+10
As a current bass player and former professional sound engineer, I can't begin to number the instances of when I had the FOH balanced well and then the bass player turns up a little bit. I turn him down a little bit in FOH to rebalance - when prompts him to turn up more... until the FOH is no longer in control of BG levels. After that point, you're mixing everything else to blend in with whatever the bass guitar guy is doing... plus usually fighting to allow the vocalists to still be able to hear the monitors.
Being a player now, I'm constantly reminding myself that what I'm hearing on stage is not the FOH balance or tone.
Like it or not, the FOH engineer ends up being the defacto 'producer' for the sound of a live event. Good ones will be able to accept input from an actual producer regarding the desired end tone result and work to reproduce that. Others will just go with whatever 'sounds right to them'. The goal of any amplification on stage should be only what is needed onstage (assuming a FOH/venue size large enough that the individual instrument amps aren't needed to provide a significant contribution to the FOH mix.)

