It would be interesting to see if toggling /org/gnome/software/allow-updates
includes update checking for extensions as well.
edit: related topics:
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Add option to disable extension updates (#2514) · Issues · GNOME / gnome-shell · GitLab
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No way to disable extension update checking (!1099) · Merge requests · GNOME / gnome-shell · GitLab
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No way to disable extension update checking (!1099) · Merge requests · GNOME / gnome-shell · GitLab
So, 2 years ago someone was already trying to introduce a switch for turning extension updates ON/OFF. That merge request is still open.