If you have interesting examples of such a relatively positive use of "check your privilege", I'd like to see them.
My experience is the same as Lumifer's - I have only seen this phrase used to shut down unwanted opinions or unwanted participants. Theoretically, it could stand for what you said, and I'd love it if it did, but in practice it doesn't seem to happen.
(Interestingly, the same seems to be true about the obnoxious -splaining family: "mansplaining", "cissplaining" etc. That is, I can well imagine their uses that, while rude, seem somewhat justified. But I don't think I've ever actually observed such a justified use; all the uses I've seen were always as a way to attack an opinion based on race/sex/identity of whoever offered it).
FWIW, in my social circle it's often used in the first person. As in, "my first response was to dismiss X as completely unnecessary; then I checked my privilege and reconsidered what X might offer to groups G1, G2, and G3." I don't necessarily claim that these sorts of uses are interesting or positive (that's a discussion I don't choose to get into here), but I don't quite see how it involves shutting anyone down.
As for "-splaining", I more often see it used as a way to attack a conversational strategy than directly to attack an opinion...
Now, it is said we all here pride ourselves on our intelligence, rationality, and moral sense. It is also said, however, that we are a fiercely independent bunch, and that we can let this pride of ours get the better of us. There have also been comments that the live communities that appear at meetups provide much more positive interactions than what goes on on this site's discussions; this might merit further investigation.
My point is; we've done a lot of research on how to do proper ethical and metaethical calculations, and on how to achieve self-empowerment and deal with our own akrasia, which is awesome. We've also done some work on matters of gender equality, which is very positive as well. But I haven't seen us do anything about the basic details of human interaction, what one would call "politeness" and "basic human decency". And I think it might be useful if we started tackling these, for our own sakes, that of those who surround us, and that of easing our mission along, which is, as I understand it so far, to save the world (from existential risk (at the hands of (unfriendly and self-modifying) artificial intelligence))).
What inspired me to propose this post was a video I just saw from Hank Green of the famed and fabled vlogbrothers. I hold these two individuals in very high esteem, and I would expect many here to share my feelings about them, on account of their values and sensibilities largely overlapping with ours; namely the sense that intelligence, knowledge and curiosity are awesome, and that intellectuals ought to use their power to help improve themselves and the world around them.
Here it is; I hope you enjoy it