The apparent compassion within the phrase is merely a thin veneer. Strip it away, and you are left with a statement saturated in resentment. The phrase is not a call for compassion, but an accusation - one demanding the accused immediately shut their mouths and keep it shut.
In an ideal world, the phrase "check your privilege" would immediately be countered with something like "check your resentment", but this is not an ideal world, and I very much doubt Leftists could ever understand the phrase "check your resentment" if they were ever confronted by it. There is no such thing as resentment in Leftism, for the Leftist interprets resentment as nothing more than a rational call for justice.
Getting Leftists to acknowledge their intrinsic resentment is probably a futile endeavor, and it seems pointless to even try. A more fruitful and necessary undertaking for those of us opposed to Leftism should be to focus on ridding ourselves of any and all resentment. This does not imply we should never feel displeased, annoyed, or outraged, but we should express these feelings with strength, nobility, and dignity untinged by weakness, vulgarity or inferiority.
We must oppose the Left. Nevertheless, we should avoid wielding the same sword the Left employs. The Left's resentment is fermented spite, contempt, and envy, and their compulsive reactions are the results of long periods of bitter contemplation. We who oppose the Left should aim to display a more dynamic confidence and fullness of being, one free of envy and malice. When we react against threats to our well-being, we must ensure our being remains grounded in and supported by ultimate Being and Reality.
In addition, we should never indulge in despair. Regardless of how hopeless and futile circumstances in the world appear. When we lose hope, even for a moment, we become cowards, and we lose that grounding and support in ultimate Being and Reality. If we lose that, we are nothing.
When the Left demands we check our privilege, we should concentrate instead on checking our resentment and despair. If we possess either, we will defeat ourselves long before the Left ever could.
Note: This post is more a note to self than it is general advice. A self-reminder, if you will.