If you are just looking to repurpose an old device for around the house use and it won’t ever be leaving your home network, then the simplest method is to set a static IP address on the device and leave the default gateway empty. That will prevent it from reaching anything other than the local subnet.

If you have multiple subnets that the device needs to access you will need a proper firewall. Make sure that the device has a DHCP reservation or a static IP and then block outgoing traffic to the WAN from that IP while still allowing traffic to your local subnets.

If it is a phone who knows what that modem might be doing if there isn’t a hardware switch for it. You can’t expect much privacy when that modem is active. But like the other poster mentiond a private DNS server that only has records from your local services would at least prevent apps from reaching out as long as they aren’t smart enough to fall back to an IP address if DNS fails.

A VPN for your phone with firewall rules on your router that prevent your VPN clients from reaching the WAN would hopefully prevent any sort of fallback like that.