Handshake domains are only accessible by a tiny percent of people, make stuff like HTTPS very difficult[1], and no matter how devices eventually use Handshake, you'll always need to have a domain on a normal TLD because there will always be devices (like TVs and old phones) that will not support it.
And what benefit do you get anyways? A custom TLD? There's already so many new TLDs but most domains are on gTLDs or ccTLDs because thats what people recognize. Even Google and Apple barely use theirs. Ownership? Not really. Handshake only manages TLDs. Buying a subdomain (like you can on Namecheap) doesn't happen on the blockchain, the owner of the TLD can take it away anytime. Say what you want to say about ICANN, but they do have rules (such as contingency plans) that new TLD owners have to follow. In what world is buying a handshake subdomain from an unknown person beholden to nobody better in any way?
[1] https://www.namecheap.com/support/knowledgebase/article.aspx...