Hello Corentin Maillard.
Welcome to Microsoft Community.
Regarding to your information I assume that you have some issues when you use the add text feature.
We can't reproduce your issue but it seems that some users have the same questions as you.
So it's may the Edge Profiles issue.
First, you can try to access this website and create a new profile, open the PDF in the new profile (Copy the address to the new profile address bar)
Check whether this feature works in the new profile.
If it can't work please press Alt+Shift+I to submit your feedback about this issue to the Edge development teams and try to use the
Canary Version Edge and check whether it works well in Canary Version Edge.
Become a Microsoft Edge Insider | Microsoft Edge
If it works well in the new profile you can use the new profile for PDF or you can sign out of your account from the old profile and login it in new profile
to sync your data to your new profile.
If there is anything not clear, please do not hesitate to let me know.
Best Regards,
Tommy-MSFT | Microsoft Community Support Specialist