I generally don't like when technology gets itself in the way to solve a problem. Why would you need the user to make your search engine the default one before even trying it? This is red flag at least for me, far from convenient and I doubt that installing an extension can be justified by privacy. If your search has better value to me than say, Google, why don't let me make the decision of putting it as my default?…
Edit: Ah, now I see. It's because I'm a Safari user. If you go to you.com in Safari it works fine. However, if you go to it in Chrome it does indeed work as you described, asking you to install the extension and whatnot. My mistake.