If your preferred search is not on your list or indexed automatically, you can add it manually. Navigate to Manage Search Engines as you did above and click the Add button.
There is no “Add” button except for the site search one. Editing an existing engine greys out the web address so it forces you to use the ones listed. There is no other obvious way to add a new search engine. Adding it as a site search and then activating it as a shortcut does not work when searching in the address bar. This is microsoft tier behavior.
My default search is Google. So obviously if I went to type something in, it would show Google. We’ll do Testing 123. Notice the G on the left side? That shows it’s searching Google.
But on Site Search, we see I have !n for Neeva. So let’s use that shortcut.
When I type !n and hit space, it turns into Search Neeva and I can add my search term:
Of course, making Brave Search, Google, or any other search engine my default will put it back on the Site Search list.
You can delete existing search engines. And if you want your own custom stuff, can add it via Site Search below. Nothing stops us from creating duplicates.