It should also be noted that Burnham goes through a lot more in two seasons (Captain and important person in her life dies partly through her own fault, she's stripped of rank and blamed for a war, her boyfriend turned out to be a murdering Klingon spy, her brother is accused of murder, Saru almost dies, she briefly finds her mother and loses her all over again and gives up literally everything in her life except for her bonds aboard Discovery to save the univrse) than Picard, who off the top of my head gets assimilated, loses his brother and nephew and whatever was going on in "The Inner Light".
If you mind Discovery's high density of events that would put one in emotional turmoil, that's fair enough. From my own viewing, I never thought that Burnham reacted unexpectedly or overly emotional to any of that, and I think that a bunch of criticism that Burnham gets for crying may be directed at a symptom, rather than at the core issue (of Discovery consisting of many events that can reasonably result in emotional turmoil for Burnham).