I pretty much know it's not possible in the Admin UI, though that would be a welcome change.
Is it possible to use the Fluent API to set the template of a page?
This is actually possible though the UI as we provide the batch action to change the template through the top menu by checking on the pages you want to change and going into More actions -> Change Template from the top navigation. You can see the attached image for screenshot.
The one specific here is that it places this change in draft mode as it generally calls for a review if your change has gone through as expected. You can of course batch publish using the same menu but I would advise on checking if widgets are placed in the expected placeholder prior to going live.
Please let us know if this works for you and if you have any further questions on this topic!
I was wondering, to help speed up the process of applying a template to multiple pages, is there a way to show a new column on Pages that lists the template in use by each page? Right now, we have mix of templates in use at the same level in the pages hierarchy.