QuickTime X is integrated part of the system software, and not available as standalone installer.
There are a few general troubleshooting tips to try, that may help pinpoint where the issue lies.
How to test an issue in another user account on your Mac - Apple Support
This will tell if an issue is due to settings in the user space or the whole system. This will help where to look.
Use safe mode to isolate issues with your Mac - Apple Support
Safe Mode temporarily turns off some non-essential components and flushes some caches. This will help where to further troubleshoot.
macOS: How to troubleshoot a software issue - Apple Support
If the issue is confined to the user space (normal user account), and safe mode doesn’t change it, then it may be due to a ‘corrupt’ preference file. There are many preference files and supporting files associated with QuickTime, but most can be re-created by removing the old ones (move to an inactive place; not the Trash just yet). One should also restart directly after moving said files, as they are technically in use at that time.