Apple TV setup

No, you cannot stream from one Apple TV to another and yes, you were given correct advice by Apple unless of course you want to physically carry the one Apple TV to the room you want to connect to.

I have 4 Apple TV's to support the 4 TVs in the house.

BTW, there are some resellers that have them on-sale right now. Scroll to the How to Buy section of https://www.macrumors.com/roundup/apple-tv/ and you will find them.

Devices you can use with the Apple TV app and Apple TV+

https://support.apple.com/en-ca/HT210740

A good strong network at home and a HDMI cable. Get a good quality hi speed HDMI cable such as

https://www.bestbuy.com/site/rocketfish-4-4k-ultrahd-hdr-in-wall-rated-hdmi-cable-black/3720002.p?skuId=3720002