I'm not so much saying we know much about them all per se, Burnham has had the sort of focus no character has in trek (again, possibly except Data). However they are convincingly fallible human beings (or alien proxies thereof). I don't expect them to necessarily always do the right thing, to shine as slightly different flavours of perfection. I look at how they present on screen and think "these people might let me down and I'd forgive them the way I do people in the real world, because that's what they are, real". I might understand how and why they let me down, empathise with those failings and be all the more inclined to celebrate their successes as a result.
For example when I look at TNG I see Riker, he is suave, bold, classically heroic. He was those things last week, he will be them next week. Those qualities are just words, they don't mean I actually believe in him, I just know he'll save the day with a flourish and we'll get recurring hints about his romance with Troi. He'll do whatever bold, suave things are required of him in the episode without any sense of the motivations of the person behind them. If he stays the same from week to week I'm happy, it's safe and predicable. When he acts in ways I don't expect it's by and large down to exactly the sort of inconsistent writing we talked about earlier and will have no bearing on who he is next week.
When I see Lorca, I see a person I might meet in real life, someone I'm trying to figure out as we go along, someone I'm never quite sure about. He surprises me, does things that impress me, he does things that make me angry. I don't trust him, but I know he might be able to persuade me anyway. I don't expect him to necessarily always be there next week miraculously unchanged by this weeks adventures or act in whatever way neatly brings us to pressing the reset button. I feel hurt, betrayed and let down that he so viciously beat that man and insulted his sisters memory after I had been willing to put some faith in him.