I hear something flying just overhead, but I can’t see anything!
One more thing before I run off downtown, then I promise I’ll shut up for a while.
Please don’t think I’m disrespecting your idea, because I think the main thrust of it is good. What we want, as Boyo Jim alluded to earlier, is to have confidence that the design firm is producing a quality product. That’s what it boils down to, IMO; we want to be able to drive across that river bridge without the fear that we’re taking our lives in our hands when we do so. I hear that.
We could do one of several things to accomplish that. One alternative, to build off of your idea of the inspectors doing it, would be to hire an independent consultant to check the calculations & design work before the project is constructed. I think that’s a great topic for debate. I can see pros and cons already. 
Another alternative would be to require the design firms doing MnDOT work to participate in a Peer Review program. Such a program would have other engineers going to the design firm to check out their process - everything from accounting to quality control. If the review folks see any deficiencies, they will note them in their report. MnDOT could require a certain number of passing reviews over a certain period of time, in order to do business with them.
I kind of like Peer Review because it helps to promote a climate of quality in design firms. Instead of nitpicking over each project coming into MnDOT, the firms themselves would be encouraged to practice good quality control, which would reduce the possibiliy of errors across all projects.