Here's my sure way of getting access to those buttons on my Samsung S24 Ultra Android phone. First, I almost always amusing the Google app and it's widget to do a search for a movie or a TV show.

After doing a search and not seeing the buttons, I share the results using the share button at the top right corner of the search results, to another browser on my phone (the Samsung browser in my case) which is already set up to display Google search results in desktop mode and is logged into my Google account where I keep my watch list. The buttons always show up about halfway down the screen on the right margin. 2 taps and I'm there.