Don't think I am ever going to allow DI again. Shouldn't they have put a mic on the cab instead?
There are times when running bass through the PA is a huge improvement. However acoustics, the quality of the PA, and the skill and artistic vision of the audio tech are significant mitigating factors.
Your stage volume in relation to the level of the PA are also significant. In order to do a good job, the PA must dominant the room and be capable of focusing direct sound on the audience area. It's very common for the stage wash to be so loud that the PA cannot dominate the room. In fact, sometimes the best thing the audio tech can do it take the bass completely out of the system. It's a mistake to assume this is a recipe for great bass tone.
Keep in mind the stage wash may consist of sound from your bass amp as well as bass going through various monitors. As an audio tech, I frequently pull the bass out of the subs and mains. But I don't typically pull bass out of the monitors without the ability to negotiate the change with the talent.
When the stage wash is excessive (during sound check), I will discuss the problem with the talent and ask the group leader to come out to the mix station and listen. This is the so the group leader can choose to improve the house sound by reducing how much bass is going through the monitors. IME, usually the group leader will reject the idea of turning down the bass on stage.
When bass is put in the PA, under most circumstances using a DI is the best option. There are exceptions where mic'ing a cab is a better choice. My impression is many audio techs prefer to avoid a mic. It's frequently not your call.
In regards to having good tone and control over it: There is a pretty significant disconnect between what we want to happen and what actually happens. This may seem offensive, but it may actually be good advice to
Lower you expectations.