With that sort of statistics, you get the feeling they deny a certain number of claims for no reason, with the hope that you won't appeal and just pay. I have little faith in health insurance companies.

Actually I did once meet someone who had worked at a health insurance company and said that they told employees to just skip every fourth claim and deny it without looking at it, exactly as you describe: in the hopes that it just wouldn't matter enough to some of those people, and they wouldn't appeal.