In Iowa, there is nothing like the MoT inspection required. There used to be. They got rid of that when I was a kid. I remember the sticker you had to put in the corner of your windscreen every year. Then they went to requiring inspection only when you sold the car. A decade later, they got rid of it completely. I remember as a pre-teen looking through the newspaper classifieds (remember them?) at used cars. The seller would list them as "red title" if the car couldn't pass inspection. One used to be able to pick such cars up for a little bit of nothing. We have never had any kind of emissions test. I am completely ignorant of what the other 49 states do or do not require.
Expanding on the general subject, and again this is for the State of Iowa, if a vehicle sustains damage that exceeds 70% of its fair market value before the damage, then it is given a Salvage title, and cannot legally be operated on public roads. If one repairs such a vehicle, at that point it must be inspected. If it passes inspection, the vehicle will be given a Rebuilt title (which, up until a few years ago, they used to call a Prior Salvage title).