In a typical studio environment with typical interior lighting, with camera settings of, say, ISO100, f/8 and 1/125 SS and without firing the flash, you will get a black frame, ie no ambient recorded. With flash duration usually being way shorter than the shutter speed, the latter doesn't have any effect on the exposure and the flash will generally freeze movement (up to a point, as flash duration increases in proportion to flash power output).
As bobby350z says, in the studio, higher shutter speeds are only really needed for freezing fast moving objects or action. As for "bright strobes", just turn the flash power down if the exposure is too hot. So, yes, perfectly possible to shoot at f/2.8. If a wide aperture like this is starting to let ambient affect the exposure, then turn off some or all of the interior lights and work using just the modelling lights.
When balancing flash and ambient outside, the whole thing becomes trickier as much faster shutter speeds will be needed to keep the ambient exposure where you want it, especially with wider apertures. Although leaf shutters allow faster sync'd shutter speeds, it's not a free lunch. If, say, you are shooting at 1/1000 to get the ambient where you want it, but flash duration for the required flash output is 1/500 (for example, because the flash is being used at full power output), the shutter will be closed before the flash has fully discharged, resulting in the desired balance of ambient and flash not being achieved.
In my own tests (in studio, no ambient light) using an X2D with XCD 38V lens and a Profoto D2 500ws flash at full power, exposure starts falling off around 1/1000 SS. This means that at full power, the D2's flash duration is longer than 1/1000.
