The website is for Apple TV+ (and MLS) subscription content only.
Store purchases require the actual (full) Apple TV app, which should be more user-friendly in most cases. Your MacBook Air M2 should have the Apple TV app for this. Older computers can use the iTunes application for Store content.
The Apple TV app is available on many devices: iPhone, iPad, iPod touch, Mac, Apple TV box, Apple Vision Pro, Windows 10/11 PCs, smart TVs, third party HDMI streaming devices. [Select models third party devices may have a somewhat limited Apple TV+ app (note the different name), that doesn’t have a Store or Library.]
How to watch Apple TV+, MLS Season Pass, and more - Apple Support
Get the Apple TV app or Apple TV+ app on your smart TV or streaming device - Apple Support
Watch on these smart TVs and third party streaming devices - Apple Support