This is so simple.
If:-
a) The amp provides a tone that is integral to your sound
b) You use many or varied fx to get your sound
c) You color your sound with anything AT ALL to make it right for you for a song
Then the FOH needs that to properly reproduce that sound in FOH.
If that sound:-
a) requires the cabs' interaction
b) requires any other part of the signal chain after the DI in the head
Then the FOH needs a mic to properly reproduce that sound in FOH
Else IF:-
a) You rely solely on your hands/bass/strings/ebow to get your sound
b) No part of the signal chain from your bass to your ears on stage in your opinion MAKES your sound what it needs to be
Then the FOH will be able to get your sound solely with a DI after the bass
Does anyone here really believe that say, Doug Wimbish or Bootsy or Tony Levin actually run pre eq DI from the bass?
Cos they are using so many fx in there chain as part of their signature sound that that eq is almost nothing compared to the other stuff.
I've done extensive FOH, monitor mixes and been the bass player, and been an engineer in the studio. And it boils down like this, FOH cant recreate any of that good stuff at all, its got mic pres, parametric eq, compression and additive fx (reverbs & chorus etc) but none of that will typically help build the sound of an SVT, or a mutron or whatever.
So if you need that colouration on the sound of your playing to make it right for you/yourband/that song/thatchours/that lick, then FOH need it in the signal they get.
And dont give me all the "FOH cant deal with it, the spillage is to much into the bass mic" nonsense. They can deal if they have any real knowledge, if the spill really is so awful build an iso box with a miced cab in it, but typically it isnt since the people who depend on that tonal coloring are rarely playing coffe shop smooth jazz through a vocal PA (in which case it wouldn't matter anyway due to venue size).
I'm absolutely with JimmyM on this one, believe you me it is very very rare to get a really bad DI out these days unlees somthing is broken or plugged in wrong, eq'ed or not, and if the player uses extensive fx or eq changes or amp colouration then it is usually IMO & IME easier to mic the cab to get it. The FOH guy is going to compress the cr@p out of you anyway to control peaks, he'll most likely limit everything to save heis big bins anyway, he will eq you to frequency mix you in with the rest of the band etc etc, he will not ride eqs (on the bass) through the gig, so if you are going to change stuff alot let him know first, he is not going to add and remove fx to make it funkier - that is your job, he is not going to provide cab or amp colouration if you need it let him know, he will reach for a mic.
He is going to take the sound you need and get it out front as best he can. He is going to make the band sound great if possible, he is definitely going to make the Kick drum sound like a bomb going off and feel like a punch in the chest, he is definitely going to make the vocals very very loud, and the guitar too probably unlees he knows your band's sound is very different from that. Give him what you want the audience to hear and within the bounds of the mix he will give that to the audience.
Lastly neither method is right all the time, it is in fact down to the bassist/band in fact and not the sound guy at all. He should have a huge array of tools to tame whatever you give him to make it right for the venue, notch filters, dynamic processors, eq etc, and if he needs to he will use them. The one thing that will really get up his nose is too much bass on stage, annd a disrespectful band. Other than that he will attempt to accomodate you as best he can.
Sorry this is such a long rant but professional sound guys deserve more credit than they seem to get here for understanding signal chain requirements for a truly excellent bass sound IMO & IME, and if they dont they shouldnt be working in the business and dont use them again!