You're right about the illogical nature of much of mental health and I can understand your difficulty with this. This illogicality is what makes mental health so difficult for outsiders to understand I think, partly anyway. Two things about this though. Firstly, from personal experience I do accept the illogical nature of my difficulties and it is something that I struggle with as someone with mh difficulties. I am in myself a very logical person but for some reason logic just does not always work when it comes to a sick mind. I think if you could work with people who have this acceptance / insight perhaps this would be easier for you. Being a support worker in my opinion is about 'being there', not necessarily understanding that illogicality or offering solutions. One of my biggest lessons with anorexia, and hardest things to come to terms with, is that change ha to come from within the individual. This is really hard for outsiders though, I am now seeing as I'd like to help people in the depths of anorexia. However, sadly, unless the individual makes those changes, takes those tiny steps towards better health themselves, then none can help I believe.
In terms of anxiety about 'clicking' with someone, I think that is completely natural and if ever you were to go on to do training to make this a profession then I would expect you would be supported with this. However, thinking about working on a more casual basis with it, I think it would be fine for you to be 'picky' about who you could / couldn't meet with and you would have the right to stipulate this. At the end of the day in any relationship, it has to work both ways, and in my experience this is crucial with support workers / therapists etc too.
Sounds like you're thinking really carefully about it which is brilliant. It's so important to get it right and not let yourself be pulled back. Take it easy! Message me if you want at all. When I get the privilege to 'inbox' people I can reply or email if you'd prefer.