

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.
All of the “snooping” is self contained. You run the network controller either locally on a PC, or on one of their dedicated pieces of hardware (dream machine/cloud key).
All of the devices connect directly to your network controller, no cloud connections. You can have devices outside of your network connected to your network controller (layer 3 adoption), but that requires port forwarding so again it is a direct connection to you.
You can enable cloud access to your network controller’s admin interface which appears to be some sort of reverse tunnel (no port forwarding needed), but it is not required. It does come in handy though.
As far as what “snooping” there is, there is basic client tracking (what IP/mac/hostnames) to show what is connected to your network. The firewall can track basics like bandwidth/throughout, and you can enable deep packet inspection which classifies internet destinations (streaming/Amazon/Netflix sort of categories). I don’t think that classification reaches out to the internet but that probably needs to be confirmed.
All of their devices have an SSH service which you can login to and you have pretty wide access to look around the system. Who knows what the binaries are doing though.
I know some of their WISP (AirMAX) hardware for long distance links has automatic crash reporting built in which is opt out. There is a pop up to let you know when you first login. No mention of that on the normal Unifi hardware, but they might have it running in the background.
I really like their APs and having your entire network in the network controller is really nice for visibility but my preference is to build my own firewall that I have more control over and then Unifi APs for wireless. If I were concerned about the APs giving out data, I know I could cut that off at the firewall easily.
A lot of the Unifi APs can have OpenWRT flashed on them, but the latest Wifi7 APs might be too locked down.