You are certainly following the correct process to export your Pages document as a Word file. Unless you are missing a screenshot, among the Share sheet options should be Save to Files.
You best approach to resolving the problem would be to Offload the Pages App - perform a restart of your iPad - and then re-install the Pages App.
To offload the Pages App:
Settings > General > iPad Storage - scroll down the list of installed Apps to Pages, tap, then select Offload App
Close-out the Settings App, then force-restart your iPad:
Force restart iPad (models with Face ID or with Touch ID in the top button)
If your iPad doesn’t have the Home button, do the following:
- Press and quickly release the volume button nearest to the top button.
- Press and quickly release the volume button farthest from the top button.
- Press and hold the top button.
- When the Apple logo appears, release the top button.

Force restart iPad (models with the Home button)
Press and hold the top button and the Home button at the same time. When the Apple logo appears, release both buttons.

After restarting your iPad, re-install pages from the App Store:
https://apps.apple.com/app/pages/id361309726