Like Millsart says, pretty much any m4/3 camera focuses fast. Very fast actually. Most of them also have a touchscreen which makes the whole shooting experience even more responsive. You can touch the screen anywhere to move the focus point or even trigger the shutter at the same time.
Okay, well I just got an EM5. Now I need the 12-40 zoom... I'm a **little* over budget, but think the AF and other aspects will make me happy. And I get to have a viewfinder.
EM5 has a pretty decent EVF in terms of brightness and refresh, however, keep in mind that its a fair bit lower in resolution than cameras like the EM1, Fuji X series, current Sony's etc
EM5 is about, what, 2.5 years old, so bear in mind the EVF does reflect that. Coming from the X100 (not S) it shouldn't look like a downgrade
If EVF is important though, the newer offering such as GX7, EM1, A6000, etc, with the 2+ million dot displays are considerably sharper, especially for things like MF
Otherwise EM5 is a pretty fantastic camera for the current used prices.
I'd suggest picking up the horizontal grip though, makes the camera way more comfortable to hold. I think there are even some cheap Chinese knockoffs these days.
If I were to go back to a m4/3, I would probably go pick up a GX7, especially in black. It's got the RF form factor, which after shooting so many cameras over the past few years I decided it's the one that's most appealing for me. Shooting the a7 like I do now, it makes me wish for a NEX7 style FE mount camera.
Reason that I would get the GX7 over a EM5 or EM10 is that it's smaller, lighter. It has the 1/8000 shutter that the EM5 doesn't, it has the fully electronic quiet shutter (but not without issues), it has one of the best focus peaking implementations, it has a really nice magnified view, and full touch screen (even able to use touch screen on the menus).
I'm a **little* over budget, but think the AF and other aspects will make me happy. And I get to have a viewfinder. 