Separation between player knowledge and character knowledge is important for fair play and avoiding power-gaming, so if players are refusing to accept the in-game penalties that come with a character being upset or their character is making decisions that are inconsistent with the character's knowledge/age/alignment, that's definitely a problem. But roleplaying out scenes in-character doesn't inherently hurt or help the game. It's all about what the players want out of the experience. Phrased as a question about whether character distress 'should' be role-played out, there is no good answer to this. It is dependent on what your roleplaying group wants out of rpgs.
There's no right way to play; some people are more into the role-playing, immersive story thing than others. If you want to roleplay things out and the rest of the group doesn't, you might want to try playing with a different group.
However, the second half of your question, 'how can I encourage more roleplaying' is much more answerable. Including mechanical consequences for characters' fear or guilt is definitely one way of doing it, and Edbury's suggestions for systems that do that more naturally than D&D are good ones (I can think of others if you want a longer list). Although it's not exactly the same as your question, I think you'll find the answers herehere have many good ideas for increasing game immersion and making players shoulder consequences for their actions.