Don't ignore the doubt
There's a lot of unspoken assumptions in societies about how things work. In democracies, at least the US one in which I was raised, there was a strongly held belief that somehow voting picks the best people for the job.
Even as early as middle school, it was pretty clear that wasn't true. One just needed to look at a middle school student government election: who chose to run, who was elected and who should have been elected. The person who got the most votes was rarely the most competent. Often they were the most popular and sometimes they had even bought that popularity by giving out free candy to their classmates.
I never really openly questioned this belief in the power of voting because everyone seemed to believe it in so strongly. To question such a strongly held belief seemed akin to blasphemy. It's not that voting isn't useful and that democracy doesn't sometimes work. It's that there's often much more than just voting that makes the system work even when it appears as if voting is doing most of the heavy lifting. That something else could be many things: shared cultural beliefs, a less visible alternative power structure, the rules about who gets to participate in the election or rules about who gets to vote.
I bring this up not to pick on democracy or voting but because there are widely-held beliefs like this where the beliefs or the justifications for the actions don't hold up to basic scrutiny or empirical observation. Instead of ignoring that lingering feeling of doubt, we should dig deeper to try to understand the bigger picture of what's really going on.