As others have said, compatibility mode just means that the documents do not support the most recent features of the Word 2016 *.docx format. In practice, that doesn't mean much, though, as Word 2016 can still edit them.

If documents open in Protected view, you should be able to click Allow editing. Note that if the affected documents are really old you may have to change the settings for Protected view in Trust Center.

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Stefan Blom
Microsoft Word MVP

Microsoft 365 apps for business
Windows 11 Professional