I'd like to assure you that the NTFS for Mac utility stopping working after the macOS Ventura update is not unique to you. Actually, this issue happens due to the late upgrade of NTFS for Mac utilities, which offer no support for the newly coming macOS Ventura.
Meanwhile, Apple removes the /System/Library/Extensions/ntfs.kext kernel on macOS Ventura, so Mac users lose the write access to an NTFS drive. You're recommended to use iBoysoft NTFS for Mac to gain NTFS write access on macOS Ventura which offers NTFS write support to macOS Ventura as long as the new OS comes to the public.
You can free download and install iBoysoft NTFS for Mac on your device which will mount your NTFS drive in read & write mode automatically as long as you connect an NTFS drive to your Mac computer.
Good luck~
