Only Apple TV+ subscription shows can play in the browser at tv.apple.com. Other commercial Apple video content, such as buy-TV shows like ‘Law and Order’, may have an informational web page on the site, but no content playback through the browser.
Rather than relying on Chromecast, consider getting a streaming device connected to your TV, that supports the Apple TV app. Possible solutions are: Apple TV box, Roku, Fire TV, PlayStation (4, 5), Xbox (One, One S, X, S), ‘Chromecast with Google TV’-dongle, or Nvidia Shield TV box.
Also note that select modern TVs are now supporting the Apple TV app as native smart app. This includes TVs from: LG, Samsung, Vizio, Sony, TCL, Philips, Hisense, Panasonic, Roku powered TVs from various brands, Fire TV Edition powered TVs from various brands.
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
All the ways to watch in the Apple TV app - Apple Support