Operations above FL410 represent real compromises in safety over efficiency, and those compromises increase with altitude. On the other hand, turbofan aircraft experience little or no fuel savings in these operating environments, so aircraft certified to FL510 will not spend much time at these heights (Concorde excepted!). Cruising at the lower levels, an aircraft certified to be safe at FL510 would be expected to be safer at FL370 than an aircraft certified to FL370, I suppose.
The benefits for business jets operating at these levels mainly reside in direct routing, above the airliner traffic and higher weather.
Rapid decompression is more serious the smaller the aircraft and the higher the altitude. For instance, an 11 square inch hole (3.7 inch diameter) would depressurinse a Lear24 at 50,000' in 10 seconds. The same hole would take several minutes to depresurise a 727 at the same altitude. The time to react in business aircraft is much less than in airliners.
Next to consider is the efficiency of oxygen equipment at high altitude. At 34,000', where ambient pressure is only 188mm, supplemental oxygen delivered at 100% will suply at partial pressure of 101mm. Even on 100% oxygen, passengers will suffer increasingly severe hypoxia above 34,000 feet. Pilots, of course, require pressure breathing equipment much lower than this. At 41,000 feet 100% oxygen will just barely sustain consciousness in a normally healthy non-smoker, assuming the mask in in place and delivering oxygen before the cabin altitude reaches 41,000 feet.
At 51,000 feet ambient pressure is only 81mmHg. At this altitude there is 6mm more partial pressure of Oxygen in the blood than in the lungs so every breath cycle is actually removing Oxygen from the blood. Life expectancy is only about five minutes, with irreversible brain damage taking place in three or less. Physiologists are not sure exactly what happens in an explosive decompression at 51,000 feet because it is too hazardous to test or even to simulate.
Pilots with pressure breathing oxygen equipment have a slim margin for survival after an explosive decompression at 51,000 feet. The 25mHg partial pressure afforded by the pressure system would probably sustain a healthy adult long enough to initiate an emergency decent if the mask was fitted and delivering 100% Oxygen prior to decompression.
{Paraphrased from a Business and Commercial Aviation article "Decompression at 51,000 Feet"/May 1978}