The chest clip is a pre-crash positioner for the harness. In other words, it helps the harness stay in place on the shoulders before a crash. It's not entirely unheard of for chest/retainer clips to fracture or come apart in certain crashes, which is why it's so important to keep the chest clip positioned properly and the harness tightned properly.
If your U.S. harnessed child seat comes with a chest/retainer clip (and they all do), then, yes, you must use it in accordance to manufacturers' instructions. Many states have a proper use clause written into their general laws which require you to use your child seat as directed by the manufacturer. If you do not, yes, you are, technically, breaking the law.
European seats are not made with chest clips because, well, it's a pre-crash positioner and the harness does all the restraining work in a crash. I believe you also have to be able to un-fasten the harness with one hand. European harnesses tend to be made a little differently as well; most have some grippy stuff on the underside, the harness usually comes out narrower from the shell, or sometimes it's thinner in general. All of which contribute to optimal positioning, if used properly.
While importing a European child seat to the US is legal, *using* one in the US is not legal.