I have added a small custom code snippet to the functions.php in your child theme directory viat SFTP. The unchecked taxonomy terms are now excluded from the search.
The cut-off is simply because the search reaches the results limit (10 items by default). You can change those limits under the General Options -> Limits panel: https://i.imgur.com/2HoNBQk.png
For optimal performance I usually recommend keeping those limits under 150, but I think in your case even a bit lower number will do the trick.
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