It isn't a problem when you're just running software on your own computer and have no need to communicate with anyone else.
But that's not the case for domain names. It wouldn't work at all if we each had our own private little parallel universe, it defeats the whole purpose of a domain name system. We all need to agree on which names are associated with which IP addresses.
I'm not trying to promote blockchains as a one-size-fits-all universal solution for every problem. That's silly, no technology is a universal solution for every problem. Blockchains are very good at solving a specific subset of problems, and DNS names IMO is one of those. When you need everyone to agree on a particular fact and you don't want to designate some particular authority to be "in charge" of validating that fact then that's exactly what a blockchain is for.
One thing that might be nice is if there could be a standard for user IDs that would allow multiple systems to work seamlessly together.
You could have Mastodon continue to focus solely on being a completely open media aggregator and social network, but also have some other completely independent and secure private messaging system that uses the same user ID system. Then if you want to send a private message to someone who's made a Mastodon post you can use that and it "just works."
Creating a universal user ID system that would work across all of this is challenging, of course.