> Why do US people keep trying to route control of major parts of their life through a system where ..
Because sometimes you don't have a say, or you don't expect to use it. No politician ever campaigned on making the lines at the DMV move faster. But we (mostly) all agree you should be required to get a license before you drive. Most Americans drive, and few have chose to route this "major part of their life" away from the DMV.
Immigration process (as this thread illustrates) sucks. It's also not a process used by voters.
The process to apply for welfare in most parts of the US sucks, but the actual welfare is valuable. People think by making it harder, it will instead result in people who are less reliant upon it. Welfare recipients are a huge political target constantly. They're the individuals who are "actively hated" in this example, and they're entirely dependent on the system in that moment, because thats how misfortune works.
Many cities making construction permits hard to get, because local residents don't want their neighborhood changing, so they petition local politicians to make the process slower/harder/more-expensive. In this example they "hate" the new construction.
If you're not a high schooler and you're applying to a state-funded university, the process to prove you're a local resident can be surprisingly complicated. This is because it's designed for high school students and all the edge cases are optimized to avoid accidentally providing tax-subsidized "in-state" tuition to an out-of-state resident. For example, I wanted to take a for-fun class at a local university and because I didn't have a local high school to vouch for my residency, I needed to provide (among other things) 50+ pages of tax documents. It took 2 semesters (1y) to prove I lived in the state, and the minimum amount of time you need to live in-state is 1 year.
People who live their life in the "happy-path" case often don't deal with the government, and don't understand the struggle of these edge cases. Plenty of activities require the government. No way around it. Sometimes, people who think "small government" is the solution end up making those processes terrible by making it understaffed or convoluted to "avoid waste".