Are there math graduate schools without stressful tests?
No one gets a PhD in mathematics (or any other subject) without experiencing stress, period. But there are different kinds of stress. Exam stress and research stress are different, and some people respond better to one or the other. In particular, the time constraints in research are obviously much much more flexible.

As Nate Eldredge notes in the comments, pretty much all US programs have some form of qualifying exam. UChicago is a notable exception, but their first-year graduate courses are more difficult, to make up for this. (We might formulate this principle as "conservation of stress.") But the important thing is that even though students freak out over qualifying exams, these exams are merely a hoop to jump through, and are not the main focus of any PhD program. Don't get me wrong, they are a useful hoop, but once they're over...they're over. You will stop stressing over exams, and start stressing over your thesis. Typically, qualifying exams happen around year 2 of a 5+ year program--the remaining years are all about research.

To put it another way, avoiding a good PhD program because of exams would be like avoiding a good vacation because of a fear of flying: perhaps necessary if you have a true, diagnosable phobia, but otherwise, you should grit your teeth and get through it.