On iPadOS 17.6.1 my Pages documents are stored on iCloud Drive and I have no issue accessing them in Pages v14.1 for iPadOS. I have no issue with the same operating system and version on my iPhone, though I prefer the larger iPad interface.
Note that iCloud is a convenient location for file storage, but is not a backup solution. If you delete a file on it, Apple only provides an ongoing 30-day window to restore the document. If your iCloud Storage capacity is close to or fully depleted, this will cause all manner of application ills that attempt to write there.
When you are using the Pages Search field to locate documents, the keyboard has its shift key pre-selected and will capitalize the first letter of the filename for that search entry. If your document names all begin with a lower-case letter, then the search will fail unless you press that shift key to switch it off.
Nothing on earth can open Pages documents but Pages. Switching to another word processing application will strand your 3,700 Pages documents unless you open and convert each one to Word docx. That would be a sobering experience.