Can purchased movies from iTunes be watched without Wifi?

The Apple TV box doesn’t store movies on disk. It needs the network connection for input.

iTunes Store movies can be played offline only on the device on which they were downloaded (Mac, PC, iPhone, iPad). That wouldn’t help with the Apple TV box for fully offline. But you can stream locally (Home Sharing) with just a short online authorization check, e.g. if your bandwidth usage is metered.

My secondary option is to buy physical DVDs, which I'm not against but have a small child in the home that could potentially scratch them.

That’s why we have shelfs at 3'7" and higher to store physical media. 😉

I am having a similar experience my library is getting too big so I’ve opted for digital instead of physical. So here is what you do. You buy whatever digital content you want movies or tv. Then you can download them on your iTunes via pc or Mac. Either to the internal drive or like me an external hdd. You can change the download location through various steps of which I forget. I think you go to edit-preferences.then advanced- then just change iTunes media folder location to the external drive. Then just download your content which can be played offline on your computer through itunes. And then if you want to view on your tv just connect the computer/Mac via hdmi or similar cords/adapters