I'm aware of the problems with ISP-provided resolvers. I meant running your own resolver, like named, which queries the root zones itself, supports DNSSEC response authentication, etc.
In that case: local resolvers involve actually installing and configuring a recursive dns server, which isn't everyone's idea of fun, whereas 1.1/8.8.8.8 can be set up with a one line config file edit and then forgotten about.
To add to a1369209993's comment, an alternate DNS might be faster (as Cloudflare claims for 1.1.1.1), too. Or more stable than your default DNS. But for me, anyway, I made the switch after Frontier started pulling the NXDOMAIN stunt.
