Anyone can run their own DNS root, but good luck getting people to use it. And ICANN has a history of deliberately colliding with TLDs used by competing services.
What are the plebs gonna do? Go the the non existent alternative?
Browsers/OS would be reasonably easy in most circumstances, but there are embedded devices, load-balancing configurations, other esoteric uses for DNS (text records, mail etc) that'd have to be considered as well.
https://en.wikipedia.org/wiki/Alternative_DNS_root
There are others. But no, they don't become the new standard.