How to change a read-only file system to a writable file system on Mac Monterey?

I'm trying to manage system files in MacOS Monterey, I'm getting an error message saying the destination folder is a Read-only file system.

Is there an easy way to manage system files in MacOS Monterey?

Since Catalina - & reinforced in subsequent OSes to be even more strict, the boot volume is read-only.
It is absolutely read-only, insomuch as it's not even a 'real' volume as far as the user is concerned. It is a virtual volume, securely locked & using a cryptographic signature to enforce that.

If you play with it, you will break it & it will no longer boot.

Your solution is to only use User locations for any & all user transactions.

From The Eclectic Light Company - Big Sur’s Signed System Volume: added security protection

Every file on Big Sur’s System volume now has a SHA-256 cryptographic hash which is stored in the file system metadata. When data is read from the SSV, its current hash is compared with the stored hash to verify that the file hasn’t been tampered with or damaged. Those familiar with my file integrity tools will recognise that this is essentially the same technique employed by them.

Further hashing is used in the file system metadata itself, from the deepest directories up to the root node, where it’s called the seal. This ensures those hashes cover the entire volume, its data and directory structure. The seal is verified each time your Mac starts up, by the boot loader before the kernel is loaded, and during installation and update of macOS system files. If verification fails, startup is halted and the user prompted to re-install macOS before proceeding.

This article includes workarounds, but they will leave your system in an uncertain state & unless you really know why you must do this… don't.