My intuition tells me that there are quite a lot of dangerous cars on the roads in states where it's a free-for-all, so I'd be interested in the research you're referring to.
If it's just an annual emissions test (as I assume many are, but I'm not well informed), then there's unlikely to be any significant prevention of mechanical failure on the road.
You are right that some of the issue could be inspection quality, although I am talking about safety inspections and not emissions. The research that I saw before seemed to indicate that mechanical failures are rare these days and contribute to a very small number of crashes. Here is an overview with some links.
Be that as it may, I think the point still stands that safety is actually not the primary concern when it comes to how we decide what is acceptable on the roads.
As a society, we tacitly accept significant death and injury every year because making the changes to avoid said death and injury would render the roads unfit for actually transporting people at the speed and volume that society and the economy require.
There are other options, for example mass public transport that (as far as I'm aware) is much safer than road travel, but the US seems unable to to invest in it seriously for whatever reason.
"Be that as it may, I think the point still stands that safety is actually not the primary concern when it comes to how we decide what is acceptable on the roads."
It is the primary concern being the most important factor in determining regulations. It simply isn't the only factor. It's a balance between safety and utility with the goal of reasonable safety.
If it's just an annual emissions test (as I assume many are, but I'm not well informed), then there's unlikely to be any significant prevention of mechanical failure on the road.