Digital photography

Can you use much higher shutter speeds when using flash with mirrorless cameras?

If there was a modern camera like this but with up to date speed of use and high frames per second rate, I'd be all over it!...is there?

Most mirrorless: no, I don't think so. Fuji X100/S: yes, as it has a leaf shutter. Can sync up to something like 1/1000, assuming your speedlight is fast enough to fire at close to full capacity in that time. But 1/500, certainly...