Change Default Browser to One Not Listed

I want to change my default browser when I open a new tab/window to &udm=14, which is a variation of Google someone coded that doesn’t have AI, but I can’t add a site to the “Search Engines” category of the “Manage Search Engines and Site Search” page in settings. I was able to add it to the “Site Search” section but those websites can’t be made your default. I just don’t want to keep having to open a bookmark every time I open a new tab just to dodge the generative AI crap I don’t want.

Same or similar feature request:

Need custom search engine url edit option to add "&udm=14"

An extension for Google Chrome:

https://chromewebstore.google.com/detail/udm14/ffcpcoipaaccggomdlgaophbocccfapl?hl=en

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