...it seems that Safari Safe Browsing uses Google's safe browsing database.
I just found "On Feb 25, 2017, com.apple.Safari.SafeBrowsing.Service tried to establish a connection to safebrowsing.googleapis.com on port 443 (https). The request was allowed via connection alert." (and a second, similar rule) in Little Snitch, so that's correct.
I'd still love to know why the Safe Browsing Service craps out on me so often; I've got to restart now, after only 1 day.