All data is only stored locally on your computer. If you have not made an export of the settings before reinstalling Firefox, it would be difficult. Maybe there is your old Firefox user profile available. Please look at "about:profiles" if the old user profile is still there.

As Speed Dial sets great importance on data protection, the data is only available in the local Firefox user profile. If this is deleted by a new installation, the data is gone. So always keep a backup and export your settings.

The extension does not override settings without permission, and of course can't either. You are asked to change browser default search by built-in functions of Firefox. The additional search input box on the new tab page is independent from this.