The quiet shutdown saves time, fuel, and APU run time, no doubt about it. But there are pitfalls. You need to do all your flight time and fuel recording first and that will tempt you to do that while taxiing. (Bad idea.) If you have a tail pipe fire on engine shutdown, you have no way to blow it out. These are rare, I know. But I will only do a quiet shutdown if I know there is a fire department nearby. But that's me.
What follows comes from the references shown below. Where I think it helpful, I've added my own comments in blue.
[G450 Quick Reference Handbook pg. NF-4]
[G450 Aircraft Operating Manual §03-06-20] With the APU GEN ON, pausing for two (2) seconds between selecting each FUEL CONTROL switch to OFF will prevent a possible electrical break-power transfer.
CAUTION: CHECK ENGINE OIL QUANTITY BETWEEN FIVE (5) AND THIRTY (30) MINUTES AFTER SHUTDOWN. DO NOT SERVICE OIL ON A COLD ENGINE. IF IN DOUBT, RUN THE ENGINE AT IDLE FOR TEN (10) MINUTES. RECHECK OIL LEVEL TO DETERMINE IF THE ENGINE NEEDS OIL SERVICE. FAILURE TO FOLLOW THESE PROCEDURES COULD LEAD TO DAMAGE TO THE OIL PUMP.
There is more to checking the oil than just looking at the 1/6th page. See G450 Normal Procedures & Techniques / Transit Check for more about this.
Gulfstream G450 Aircraft Operating Manual, Revision 35, April 30, 2013.
Gulfstream G450 Quick Reference Handbook, GAC-AC-G450-OPS-0003, Revision 34, 18 April 2013