default-apps: Use new adwaita widgetry, drop deprecated API
GTK has deprecated the AppChooser interface and related widgets. These APIs will be gone in GTK5, so we should move away from them as soon as possible.
This MR takes the opportunity to change the design to match newer mockups while dropping deprecated APIs.
This implements everything from the "Default Apps" side of the new mockups aside from merging it with the Applications panel.