Why don't anime studios hire native English speakers? In every single anime I've watched that features a native English speaking character (usually American or British), they usually say a few lines in (heavy) Japanese-accented English before switching to full-on Japanese.

Why don't they hire a native speaker for those few lines? Or even hire a native English speaker to voice act American / British characters. I'm sure there are quite a lot of English-speakers in Japan with excellent Japanese pronunciation such that it wouldn't bother the anime's main target audience: Japanese people.
From time to time non-native Japanese speakers are employed as voice actors in Japan, but it's very unusual. A more ambitious anime production might use native speakers of other languages to voice minor roles whose only lines are not in Japanese. It's even rarer for a recurring or major character whose lines are mostly in Japanese to be voiced by someone whose not native speaker, even though the character is supposed to be a foreigner.

I think there are two main reasons for this, both from a "supply" and "demand" side of things. The first the Japanese public don't appear to be used to hearing their native language spoken in foreign accents as people in the United States and other western countries are. It not something they see in movies and TV shows, and rarely in their daily lives. Even a slight to moderate foreign accent, as you would expect from a foreigner in Japan, would be off-putting and risk being incomprehensible in parts. 

The second is that while there are unquestionably large numbers of native speakers of foreign languages in Japan, very few would be willing to work for the low wages voice actors get in Japan. There is an endless supply of native Japanese wannabe voice actors, with over 100 voice acting schools to train them, all desperate to get jobs in the industry. By comparison, the supply of would be foreign voice actors is all but non-existent. I know a number of people who moved to Japan, got jobs and became fluent in Japanese but they'd all be taking massive pay cuts to work as voice actors. In most cases hiring a native speaker of a foreign language would be much harder and probably more expensive than hiring someone born in Japan would be.

There's a slight hope that things might change. The currently airing anime series Sakura Quest has handled foreigners and foreign accents better than any anime I've seen (which admittedly isn't saying much). Their's a recurring character from a foreign country that speaks Japanese with a foreign accent voiced by an apparent foreigner (Vinary Murthy). In a couple of episodes there were Spanish tourists who were apparently voiced by native Spanish speakers as they spoke convincing Spanish and terrible Japanese. But I don't know if this is really a trend and I can't see the expectation of Japanese viewers that foreign characters speak fluent and accentless Japanese being changed so easily.