Problem: It can be very difficult to find a template if you don't exactly remember its name
Proposed solution: To the template insertion wizard (which is currently empty upon opening) add a list with favorited and related templates. Once you use the searchbar, the searchresults will replace this list.
Who would benefit: Beginning editors who do not know the common template names (like infobox/navbox even), and experienced users who keep using the same templates over and over.
More comments: Templates could have an option to favorite them and these favorites would show up in a list in the initial dialog (a first version could maybe use your watchlist data for this, instead of a separate list). Ideally, it would also show you "Related templates", which are used on pages that share categories with your article. This is similar to how WikiData's recoin works.