I also want to vote on this feature, with a quite specific additional request. (The urge is so strong that I create an account to leave the comment here.)
I basically follow the steps recommended by community members here. It works fine with a caveat: the URL of customized shortcut (could be set as default for omnibox search) cannot include port number, and I wish Brave team could consider improving on this.
My use case is, I self-host SearXNG and wish to use it as my default search engine service. In this case the address would be something like:
http://127.0.0.1:8888/search?q=%s
My experiment is that, while I could input the full address, including port number 8888 in my search engine setting, the port number would be removed after I restart Brave, so when I use this shortcut/search in omnibox, the resulting address would be:
http://127.0.0.1/search?q=%s
As the port number is removed after restarting Brave, I cannot access my self-hosted SearXNG, and I have to adjust the search engine setting every time I restart Brave to include port number again, which is far from ideal.