With your size it wouldn't be all bad to consider going to the 500s like the Ninja 500 or GS 500 Suzuki. A bit physically bigger. Don't get all wound up on seat height that entirely has to do with confidence. I am 5'6" with a 32" inseam to ground, my one bike has a 35" seat height. I can't flat foot on the bike, but it doesn't matter I have no problem with one foot down on the ball of my foot, I know I have the ability to deal with it. At 6 foot you shouldn't have much issue with anything. All that matters if the seating position is comfortable.

You also could probably ride the 650 twin from Kawasaki, the Ninja or the Versys, they'r far less a hand full than the 600 supersports. Especially the Versys. Then there is the DRz400SM with a more supermoto look, but not the comfort of those others.

I'm thinking you should do a basic rider course to get a feel for the smaller bikes and develop the basics if you haven't already done so. The instructor could also offer their opinions on your capabilities for what to ride.

 

Your first bike should always be a standard configuration, and a cheap one.
The problem with most smaller bikes is that they are made for smaller sized folks.

I'm 6 ft and have never owned a bike under 1000cc that I didn't have to fit to me.

Back in the day when they made flat square two up seats. You could stretch out onto the back part of the seat. Now with form fitted seats for 5ft 10 or smaller folks. You end up cramped up.

For my Sportster, I extended the foot controls, for my Hyosung I modified the seat, and took out the cockpit saddle.

BTW Hyosung makes some nice 250-650 standards and cruisers.

Both cruisers and sportbikes have some loos of control due to the way you sit on them.
Just the nature of the beast. until you get used to them.

Im picking up a 1980 Yammy XS650 for a song soon. I'll be making a custom fit cafe seat just for my fat ass.

 

I'm short, but I have a link you might want to try. It is a page where you input your height and inseam. Then you select the bikes that you want to see "yourself" sitting on. It will show you a side profile of how you sit on the bike. You can see yourself in a riding position and with your feet on the ground.
http://cycle-ergo.com/

 

It ain't about the height of the rider

It is not how tall you, but how long your legs are! I got some stubby ones, 28 inseam, but I am 5'10". the seat height will be most important for you. But also your riding style will be important. Sports bike, cruiser or touring?

I got on a buddy's BMW and was on my tip toes! I remembered my BMW in Germany a R60, I would be flat footed and knees bent.