I'm brand new to the zero turn world and read that when it's cold the Kohler motor tent to not start. If this happens can you move the mower without it running, meaning to the other side of the garage.
If the engine has a choke, use it when first starting, or if the Air temperature is cool.
My cub cadet has the Courage 19, it's starts like a top even when it's 40F outside...if course in Texas days of 40 degrees are few and far between...
But yes, as long as it is neutral, or the handles are up i believe it should roll.
Read your manual, on a Z, you will have to push in or pull out 2 pins in back behind the hydro trannys, to let the hydros roll. Hydros have no neutral, just stop, Go, & park
On my 18 hp on my gravely, I need to move throttle all the way to choke spin till it fires and dies then move throttle all the way down to idle it will then start and keep running, it's been like this from day one, if you haven't already replace spark plug and air filter, this will go a long way in helping you get it running.