The most popular anime streaming site outside of Japan, Crunchyroll, hasn't had hard-subs.... in forever. You can watch any of the shows in their original Japanese or available dubs without subtitles.
I'd be willing to wager Netflix, which has a fair amount of anime, can do the same.
It looks like they now offer the option on some content, but it automatically defaults to English and you have to manually turn it off each time. That's a step in the right direction at least.
I think it just defaults to the user's OS language (iOS, browser). Fortunately once you turn it off (or set it to another language) - that setting will persist so you don't have to constantly change it back.
I flip pretty frequently between Chinese and English subs and it'll remember the last setting between episodes / shows / etc.
I'd be willing to wager Netflix, which has a fair amount of anime, can do the same.
What site(s) are you referring to?