Ensure that Pages is not running. Click on the Desktop to get the Finder's attention. Press shift+command+G to open the Go To Folder dialog, and paste the following into it:
~/Library/Caches
This will open a Finder window inside your local Library Caches folder. Look for a folder with a Pages icon on it. It will be named com.apple.iWork.Pages. Drag that specific bundle (aka folder) to the Trash. Close the Finder window, and then launch Pages again. Any change in its launch behavior? Pages will recreate the com.apple.iWork.Pages folder.
If the above does not resolve the launch issues, I would reboot into Safe Boot mode by pressing and holding the shift key until the progress bar appears. This will be a slower boot process. Once signed in, see if Pages launches differently. Then, follow with a normal reboot, and try Pages once more. Did you observe any behavioral changes during, and after the Safe boot process?
Lastly, if you are running any anti-virus product, you should remove it entirely per the vendor's instructions as it can be impacting Pages behavior. Same removal for any so-called Mac "cleaner" products which are known to be overly aggressive with needed application bundle contents. Reinstalling Pages is a waste of time, as the issue is on your Mac.