All models MacBook Pro that can run macOS Mojave 10.14, can also run macOS Catalina 10.15. That is, models Mid 2012 or later.
Macs with macOS Catalina will have the TV app (replacing part of iTunes), in which to watch Apple TV+ and other content, and will have a search field in there.
The website tv.apple.com currently has no search functionality by design, other than visually finding shows on the home page. All the Apple TV+ content is there, though. If you have suggestions for changes, then you may let Apple know your thoughts through the feedback form. Do note however that Mojave is no longer actively supported, so no changes for that version OS are expected.
macOS Mojave technical specifications - Apple Support
macOS Catalina technical specifications - Apple Support
Feedback - Apple TV App - Apple
All the ways to watch Apple TV+ - Apple Support