I think your calculation of car sizes is a bit wide. I think most cars in the US are around 6 feet wide. Even large minivans are only around 2m. Granted, the mirrors add width and some drivers add excessive dongles on the edge of that, but conscientious drivers of wide vehicles should already be accustomed to passing slow moving vehicles more carefully. In addition, in the places where there are proper dedicated cycle lanes, the lanes are painted inside a full-width shoulder, with space to spare on both the left and the right.
I found there were quite a few stretches of road in Florida, for example, where there were 3 or 4 lanes of high speed traffic going parallel to a painted cycle path in the shoulder. It wasn't the most relaxing place to ride, but it still felt safer than going down some of the highways and arterial roads in other states where there were few (if any) cycle paths, simply because the drivers had more visual aids (signage plus lines) to tell them what to do when they saw a cyclist. I felt more confident cycling even in other parts of Florida without the cycle lanes because despite the drivers being fairly inconsiderate, at least they didn't visibly panic when they saw me there.
It is important to compare like with like, though. A "cycle path" which is just a bit of different-colored paint in the gutter of an existing traffic lane is pretty much useless. But if the painted cycle path is explicitly painted in what would otherwise be a full-width shoulder, I agree with other commenters here that it allows more flexibility to the cyclist than a separated lane (e.g. in turning left without needing to do a pedestrian-style hook turn) and it has the side effect of teaching drivers how to more carefully consider other traffic.
I really think a lot comes down to driver education. For example in areas where there is frequently slow farm equipment or horse and buggies on the main road, drivers have learned how to share the road, so it feels safe to cycle when there is no cycle path or even no shoulder at all. Unfortunately my experience in a lot of the US is that most drivers don't know how to deal with cyclists, so they'll ignorantly tailgate, cut you off or abuse you just for following the road rules. I'm not sure if that results in more casualties, but it certainly makes the experience of cycling more fraught, which means less people will want to cycle, so there is a vicious circle of increasing driver ignorance. I think painted lines at least provide some antidote to that.