The onboard flashes are pretty worthless...
They are in the worst possible place for redeye and ugly shadows (close to and right above the lens axis). They also are weak, slow to recycle and rapidly drain the camera's batteries.
For those reasons, I hardly ever use a built in flash. (On one of my 7Ds the built in flash has been stuck since new.... now 4.5 years old with about 100,000 clicks on it, I've never fixed the flash even though it's easy DIY).
So, if you ask me, don't let the lack of a horrible little built-in flash stop you from buying an otherwise very nice camera.
Just get an accessory flash.... Though I suggest also spending a little more to also get an off-camera shoe cord and a flash bracket, to get even better results from it.
OTOH, not sure what you are expecting from the "upgrade".
Unless you regularly make big prints.... really big prints, probably larger than 16x24"... you aren't going to see a great deal of benefit from going from 60D cropper to 6D FF camera. You won't see much difference in 8x10s and 11x14s... and certainly nothing noticeable for any online image uses.
Another reason you might want a FF is if you shoot a lot of low light/high ISO images. The 6D is probably good for another one or two stops high ISO. The 6D's AF system is able to focus in exceptonally low light, too... Center point only is good to -3EV (moonlight)... the other points are same as 60D's at -1EV. 6D has 11 AF points, only the center one is the better "dual axis/cross" type. 60D has 9 AF points, however all of them are better "dual axis/cross" type. 60D has that useful articulated LCD screen. 6D doesn't.
Any EF-S or third party "crop only" lenses will need to be replaced, going to FF. Plus lenses for FF tend to be larger, heavier and more expensive (especially telephotos).
But if you really have your sights set on 6D, if it were me I wouldn't worry about the lack of a flash.