Your bass amd strings and amp and speakers all color your sound and make it sound the way you want.
If you get to play by having your cab mic'd when you play through a PA ( or use a speaker level DI, then you probably get as close to the tone you like out to the audience. If you are playing direct at all, I see two instances.. 1) Bass to DI to amp.. you hear your amp the way you want, but the FOH will sound way different. You might not like it at all. 2) Bass to amp with Direct out ... then speakers. The DI can get the tone of your amp, but no coloring from the speakers. In that case, it seems that having a hifi sounding full range cab with tweaters will be more like the PA you're going through.. and that by adjusting the tone of your amp/bass you will get a similar sound on stage and off stage. In this case a tweater will help by letting you know what youll sound like to the audience.. and if you want less treble.. do it with your amp..
If you always mic your cab, you're probably golden. If you go direct.. unless you have your DI box as your amp/cab modeler, then it seems that having a hifi sounding cab with QUALITY tweeters ( not common ) would be the best option for a consistent sound. It seems that many people use their speaker choice a very large part of their sound.. but many people will never hear what you have dialed in on stage unless you are mic'd. Sound reasonable? Not trying to kick hornets nests here.
For one, it's '
"tweeter", not "tweater", but that's me just being an *******.
In general, I would agree with you, and more and more the way shows are run nowdays, it makes more and more sense to have a "PA-like" sound/speaker on stage to get a better idea of how you sound out front, make you sound better for the sound engineer who will in turn make it sound better for you, etc., etc., ...
This is where I Directly Inject my huge butt into the process....
A decent microphone on a good sounding speaker would take all that out of the process, provide natural highend and lowend rolloff, "mix slotting" if you will (provided the band is good at this on their own. A long shot indeed in some cases), and actually make their job easier.
If your band is worth a damn, and can manage to sound good on their own, then you don't need all this extra stuff. You can have a soundman, who takes your tones that already work well together, capture them all, mix levels, set mains eq to correct the room, then leave it alone, go chase girls, whatever.
If the guy has to run 5 bands worth of young people across the stage, all of whom are full of piss and vinegar and starry-eyed up and coming rockstars, yet have no clue how sound works and have their gear adjusted to sound like total crap outside their bedroom, I really pity that guy and have a huge case of empathy for him.
Some of us get upset at soundguys for playing producer instead of reproducer, but sometimes, just for the good of the public at large, they have to. Either that or drive away the entire crowd and lose their own job just for letting some band sound as crappy as they really do.