How do you cover yourself legally?

I'm getting ready to haul asphalt here in my city, and as usual the higher ups interviewing you just do the best job of stressing you out for what's to come. I have something on my record from VA back in 2019. Fender bender kind of incident, and I just paid to get out of it because I didn't have the money to pay $500 to get a Greyhound to VA for the court date, then all the lawyers fees. I just wanted out of it, so I took the easy way out. $150, and when I look at my MVR it's not under commercial, but I report it and I think these companies do take it into consideration.

Looks like I'm getting this job and I'm starting 6AM tomorrow, so I'm gonna get some rest after this. The thing that got me though is they said, "If you get a ticket just call us up." I kind of felt like doing that means I'm going to be gossiped about in the office more than get any real help, because they also said, "I think this is for most companies, but if you get a ticket it's on YOU unless we know you didn't cause it. If you move 10 feet and DOT sees you didn't have your seat belt on, we've had guys get ticketed for that. We'll try to help you along though if something happens, just give us a call."

So is it normal for these kinds of places to leave it to the driver and not defend him? When you're going to haul hazmat? When I was with another company for a bit last year, they offered to have us sign on with "US Legal" I think it was. Prime did jack of course.

What's the deal?