Apple TV second generation, 720p with Samsung 720p TV can’t stream PBS!

The 720P 2nd gen ATV is almost 13 years old and didn't use native apps (relied on server based Chanels with basic templates). The amount of 3rd party providers that have dropped support continues to grow, you would have to contact them to see if they are still compatible, and it is classified as obsolete so not much can be done for any issues that arise (which will be more common due to age).

It may be a good idea to get a cheap used 4th gen so you can use the native apps that are regularly updated along with the OS.

No problem. Yes, no matter which tv or streaming provider you will need at least the 4th Gen as that is required to have the current operating system and access to native apps via the App Store. If the TV is capable of outputting 4K and HDR then you’re better off with the 5th Gen as the previous is limited to 1080P and SDR.