Hmmm. I mean. I'm a contractor too, and I share your pain, but ... I'm really impressed you walk out of paid work because of the issue tracker your client uses.. It sounds a bit like they dodged a bigger bullet than you did tbh mate.
All these systems suck. You learn to live with them, for me I do this:
Everything goes in OmniFocus, I have a keyboard shortcut to create a task that takes 15 mins before I finish work for the day at a client, I update whatever ticket system they use (mostly Jira, but also sometimes even worse things like servicenow) and also whatever enterprise crapware my agency uses (usually some sap based bollocks).
The last 15 mins suck. But it's part of the deal. I can't imagine how strongly you feel to turn down contractor rates due to a ticket system.. I mean, come on?
Edit: Also - btw -- if you're on a Mac the app-store 'fat-app' version of Jira is about 10x better than using the web interface, I suggest you give it a try.