All babies cry. Mine had colic so she cried A LOT. But I did everything I could to comfort her and stop her from crying. I can't really understand why you wouldn't? Obviously I needed to put her down to go to the toilet or make a drink etc, but I'd pick her up again straight away and talk to her etc. I never let her to cry just because it was nighttime. That makes no sense to me.
Now she is a pre schooler, if she has a tantrum, I don't pander to what she wants (e.g. Not go to bed). But I do try to help her manage her overwhelming emotions and I stay with her, I don't ditch her on the 'naughty step' (hideous idea) and ignore her.
If she kicks off (which is rare), I sit in the room with her, explain we are not going to do X because Y, that I understand she is cross/upset/sad/whatever, and that she can have a cuddle when she is ready. Then I sit quietly and read or whatever. She usually calms down in few minutes or so, and climbs on my knee for a cuddle. At which point we talk about what we can do when we start to feel angry, and why it's not nice for anyone to get so wound up.
Seems to be working so far. Her behaviour is lovely at pre school, I've never felt the need to 'punish' for anything or leave her to cry because she's upset.
You can show empathy when they are upset and comfort them when they are crying without being a total pushover. The two aren't mutually exclusive.