Keeping one’s emotions in check is not the same as pretending they aren’t there or suppressing them. The people who, you say, “encourage and demand the full experience of whatever emotion you are going through” may be concerned that otherwise the emotions will get buried in the unconscious where they can do hidden damage.
That’s a legitimate concern, no doubt. So I agree we shouldn’t ignore the emotions. But neither should we wallow in them, encourage their persistence, or do things that end up increasing their force—as can happen, for example, when people deliberately turn on certain kinds of music that mirror and thus help sustain the “mood” they are in, or when they sit around rehearsing the conversation or mentally replaying the event that gave rise to the emotion in the first place. This is spiritually dangerous because it leads to even further confusion as to who we really are; we get lost in the maze of our feelings, forgetting that they are by their very nature in flux—”e-motion” and “movement” are related words, after all—rather than endeavoring to find that point of stability, permanence, and perfect clarity which is the goal, or rather a preliminary goal, of the whole spiritual life.
So what should we do? One tactic is to watch our emotions with as unprejudiced and as non-judgmental an attitude as we can possibly muster, neither blaming nor praising them. We can even say out loud, “I am feeling anger”, “I am feeling sadness”, “I am feeling delight”; these very propositions themselves can help us acknowledge the feeling while at the same time detaching ourselves from it. “I know I’m feeling this given emotion”, but the minute I say this I also know “I” am not the emotion.