I have seen the phrase 大快人心 used in sentences to mean everyone is happy/pleased/satisfied with something but I recently came across it being used as a stand alone exclamation. Would that translate to something like "Everyone is happy!" or "Everything is great!" ?
"大快人心" , whether in a sentence, or as a stand-alone expression, means the same thing. It has a more limited meaning than "everybody is happy or pleased". "大快人心" means that people are happy, or feel vindicated collectively, when a perceived injustice has been rectified.
For example, if your government changes a bad policy to one that everybody welcomes, people are happy and pleased, but it's NOT an occasion to say 大快人心. However, if the official, who had received tons of advice and warning against the bad policy, but who had used his power to insist on it now gets fired. Then people can say, 大快人心!
大快人心 (greatly satisfied/ pleased the people) can be translated as "everyone rejoices!" (justice is served) when used as an exclamation
If "everyone rejoices!" or "justice is served" can act as an exclamation, so can 大快人心
Any short phrase can be used as an exclamation as long as it makes sense
真是大快人心！那杀人凶手伏法了! - Everyone rejoices! That murderer is dead!