Everything behind your router is your home network, your LAN, your private network. But it seems you know this. What is the actual issue here? It seems you are keeping us in the dark about the source worry, or you found the support document that you came from to be unclear.
is it possible to have AirPlay to your home network’s devices?
It is possible to use AirPlay within your home network. All local devices using AirPlay should be on the same home network. You can’t AirPlay from another location, if that is what you mean.
The question is whether Private Addressing affects Wi-Fi connections and functions.
Home networks are characterized by private addressing for IP address. Your home network would likely bring Wi-Fi for that local network alone. What functions did you have in mind? What connections are you worried about?
The private Wi-Fi addresses in the support document relate to the “MAC address”, which identifies hardware uniquely. That MAC address would be known to the router of the Wi-Fi network (not elsewhere on the internet). A commercial Wi-Fi network (e.g. in a business) could use that as one of the markers to recognize a device from an earlier visit. Or use it for authentication: Some offices may allow on their network only devices from a list of known MAC addresses. That is bad security, but it happens.
By generating on the (mobile) devices a different MAC address for each Wi-Fi network (instead of the real static MAC address of the device), privacy is a bit improved, as devices are harder to match if compared through lists from separate networks.
The router will give (lease) a private IP address on a home network (though the DHCP function) to devices. A totally different kind of private addressing. Here private address means shielded from the rest of the internet (only connectable from the outside if you make provisions for that). 192.168.0.0/16 is such a private IP address block (most popular for home networks). If 65,534 IP addresses (“/16”) aren’t enough for your home network, then 172.16.0.0/12 (1,048,574 IP addresses) and 10.0.0.0/8 (16,777,214 IP addresses) are also possible. ;-)
My question is how do I designate my HOME Wi-Fi network as a safe home location and environment?
Recommended settings for Wi-Fi routers and access points - Apple Support
Make sure UPnP is turned off on your router. But that is more beneficial for the router owner than the guest.
The main topic addresses private addressing without any consideration or even mentioning ApplePlay, AirPlay, AirPrint and all those other things
Those things are not affected by it. Like your car doesn’t care about the color of your shirt.
