Ovinomancer
No flips for you!
I was (properly) taken to task earlier for reading "can" as "does". Onky fair to point it out, here.This makes no sense. Why give it the ability to hide in plain sight, then give it virtually no ability to hide in plain sight. Were that ability meant to be for visibility only, it would give some sort of bonus, but it fails to even give advantage. A dex bonus of 0 and no bonus to hiding for the ability means that it will be noticed the majority of the time by anyone with a bonus and the vast majority of the time by a group of people. The whole point of the ability is that it gets the jump of those it attacks. The wording in gargoyles in 5e says that they can be perfectly still by the way, and prior editions also said that they could be perfectly still. Below are quotes from 5e and 3e.
5e: "A gargoyle lurks among masonry and ruins, as still as any stone sculpture."
3e: "Gargoyles often appear to be winged stone statues, for they can perch indefinitely without moving and use this disguise to surprise their foes."
Both editions let you know that they can be perfectly still with those statements. Stone sculptures don't move a hair, and perching indefinitely without moving is also perfect stillness.
For you to be correct here, you have to ignore both logic and flat out statements to the contrary. Invisibility would only remove the ability to mistake it for a statue, not the perfect stillness and therefore perfect silence of gargoyle itself.

That said, gargoyles in 5e are indistinguishable from an inanimate statue when motionless. I think it's just fine to rule they can remain motionless at will.