Internet access is not necessary for Find My to work. Your devices all broadcast their public key over Bluetooth and NFC, and your devices all also have your own private key, and so are identifiable / visible in Find My.
Even if they hadn't had the local map, they can still get some useful signals from Find My (and they can still push Ring requests). However, in their case, their old phone still had the log of the last seen location. Again, no Internet required.
He said no service at all. When traveling, and you have an old phone with eSIM support, carrying it with you in a backpack (with eSIM information ready to go) is a good idea. Especially when traveling alone, you never know what will happen. Your primary phone can be stolen, you drop it, or it simply dies and you need rescue.
Even if they hadn't had the local map, they can still get some useful signals from Find My (and they can still push Ring requests). However, in their case, their old phone still had the log of the last seen location. Again, no Internet required.