Is your CMS optimised for touch screen devices ? Have you checked it against the mobile friendly index ?
I hope you are aware that a theme that fits itself to various screen sizes is not the only quality Google ask for touch screen devices.
Try here. You will know 😉
Most of the modern CMS support touch and related gestures. Some do it extensively using custom libraries.
I will take WordPress for granted. I saw a couple of articles where they were talking about support for multi-finger gestures and so on. ie, the website UI will behave more or less like a native app on a mobile screen.
PWA has it by default ? I think so.
Depends on which platform you use for PWA. Some have extensive touch screen support. Others have it to bare minimum.