You could be using addresses, as you can still use IPs. So no problem about privacy. It could be more like "Name of my wallet, in which all my addresses are stored". Then you would be just giving other people your wallets ID.
That is not how Bitcoin works. The network has no concept of a "wallet". The network does not know nor does it care about what addresses are in your wallet. There is no such thing as a "wallet ID" to the network (or one that is consistent across multiple wallet implementations).
Precisely, you'd have to notify the domain registrar(s) every time you wanted to use a different address, so by the time you've given them the address, it's just as easy to cut out the middleman, skip that step altogether and give the address directly to the person you would like to receive a payment from.
Addresses are mathematically linked the private key, so the "easy" way to do it (but with some pretty hefty drawbacks) would be that the domain providers themselves could automatically update the address on-the-fly if
they held the private keys and stored your funds for you. But that is absolutely
not recommended from a security standpoint. Plus other wallet software probably still wouldn't recognise the domain format, so you could only transact with others who are using the same centralised service.
I'd assume all the easy ways people could conceive to set it up would likely involve sacrificing security for convenience. Which is probably why such a service hasn't become commonplace, despite the relative ease to set it up. People generally value the security of holding their own keys over the convenience of having a domain. Plus I'm assuming someone offering such a service wouldn't be doing it out of the goodness of their own hearts and might want to charge you for the privilege.
It seems smart contracts are the safest way to implement the idea so far, but those aren't exactly newbie-friendly either, plus that currently involves using altcoins like ETH or Qora, possibly some others too (RSK, anyone? Honestly not sure on that one, never looked into it). No smart contracts exist natively in Bitcoin yet.
So yeah, generally just not as easy as it sounds.