they do not support the use of frameworks
No hosting company would under normal conditions, they won’t support the use of an application or other piece of software for that matter either. No company in their right mind is going to guarantee you support for anything. They give their specs and it’s up to you to take the time to see if those specs support the software or framework you want to use. If they do you can use the software if they don’t you can’t.
CI should run on any host with PHP support. But they won’t support you using it, that’s not their job - their job is just to provide the hosting, not to support what you’re using it for.
EDIT: I took a quick look at the first 3 hosting companies you mentioned, all of their specs fully support CI and can run it optimally.
EDIT 2: There’s also CloudIgniter, they do officially support CI.