I'm trying to make a search page which ideally would search the target assets based on their page names.
Except, it looks like I can only set a search field to look at the asset name or asset short name, not the page name. Is that right? Or has the option to search page names been added in a version later than 3.22 ?
The name of a page is stored in "Asset Name". So you'll want to search on Asset Name and potentially limit the search to just the Page asset and its child types.