I’ve already voted here, but I have another addition to why it’s a good idea for my use case at least.
When one searches using POST on startpage.com and maybe some others, searching using the search string (i.e., the url bar), the search string is saved in the resulting url [pic 1]. If you search using the search box on the website, the string you’ve searched for is not saved in the url [pictures 2 & 3]. This makes it so searching from the startpage website means searches don’t save if the tab closes down, and you can’t search something and have it load again later (the website will reload back to the home page). I like to leave searches open for later sometimes, which becomes only possible for the default search engine rather than if I searched using the website
I’m using Kagi now, but occasionally it’s useful to use startpage for its more Google-like results. This issue can be worked around by using GET rather than POST, but that complicates the URL and is overall less private/streamlined