Generally when you run right at service ceiling you are near the edge of the operating envelope with little energy margin if things go off the rails, so you are in a relatively dangerous place (also, aerodynamic forces, and damping, are drastically reduced in the thin air, but the inertial mass is the same, which tends to exaggerate overcontrolling/overshoot tendencies).
Less so if you are very light, but business aircraft are usually operating "heavy", in the first part of the trip especially, because the interior weighs so much (empty weight is permanently high) and passenger load is relatively small.
So if you are going to cruise at ceiling you have to be extra careful monitoring speed/mach to make sure you don't run out of energy margin. If you run through some smooth mountain down wave air, that starts you gently descending, to which the autopilot reacts by pitching up to keep the flight level, and you aren't paying careful attention, you might find yourself with speed decaying toward shaker in very thin air.
You have little thrust margin, and at this point, if you are sliding onto the back side of the power curve, you are in big trouble and need to descend immediately to regain energy.
So if 45000 ft is high enough to clear 99% of the weather, there is little need to go all the way to FL510 when you are much safer, from a general operational perspective, at FL450. The only motivation to go that high would to be to get clearance over the top of a cell in the tropics (mostly to avoid the hail that can come shooting out the top) without having to divert upwind of it (once you're down near the equator, even 51k may not be high enough and you still have to go around them).