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?

You're asking me upfront to grant you privileges (access to Chrome API's that are otherwise not available to you) by installing an extension? That is a no-go for many security minded people.

Also: Opening an extension link after clicking on the "search" input field looks pretty much like a dark pattern to me. This is something to avoid when it comes to a friendly UX.