How to restore previous version of iOS Video app
My answer addresses the OP issue with TV.app, not reverting to the previous Videos.app. The functionality the OP wishes to have is available in the TV.app and that is what my answer addresses.

I know this question is a little old, but I had the same issue with the new TV.app seeing my local Home Sharing library. This question is also addressed/answered here as well.

Home Sharing libraries under the library menu at the top of the TV.app do not appear until;


Home Sharing is setup on all devices. And...
Content is downloaded from the iTunes Store (e.g. a single free tv show.)


Step 2 seems to trigger the TV.app to make Home Sharing available. Once this is done your Home Sharing library will be visible in the menu and you will be able to access and view your content from the shared library on the device.

The below image is my iPad showing our local Home Sharing library iMac Library in the TV.app on iOS 10.3.1 - Hope this answer helps others.