I recently played a game which I lost disappointingly. I felt like I was playing stronger than my usual level despite being a tad unwell. The manner in which I lost has lead me to raise questions on my finishing and pressure handling ability but also the fairness of the rules of the game.
I felt like I was outplaying my higher rated opponent for parts of the game in question. I reached an endgame with decisive material and positional advantage. We were both seriously low on time. My opponent made an illegal move. The application of the rules were applied correctly but I felt the time used to apply the penalty gained him calculation time advantage. I felt distracted for the remainder of the game as my opponent played more confidently especially from this point onwards. I succumbed to time and phycological pressure (again!) and walked into a mating net so I clearly need to improve in my ways. The arbiter did all they could within the law of chess and was sympathetic towards how I felt. I couldn’t suggest my opponent was unsporting and accept the result however the question of what the appropriate penalty should be springs to my mind. It’s normally at the end of a game where tension may become most difficult to handle. 
In future would it be fairer for an instant loss to be applied for a player who makes an illegal move in a classical game when there is very little time left? A time penalty perhaps being more appropriate earlier in the game. I do believe Intentionally or unintentionally a player who makes an illegal move may get an unfair advantage to recover in a situation when they are low in time.
So far I’ve had a mixture of responses. Some believe a time penalty is appropriate. Others support the idea of awarding a win. There was a suggestion I got from an active chess playing friend where the stipulation should be that the opponent of the illegal move player should be aawarded a draw even if they go on to lose over the board. At the moment I’m favouring this more than my initial idea. I believe this idea would also be fairer in situations where an opponent is distracting or offering draws in an illegal manner.
I might not be noticing certain factors at the moment as my loss is kind of fresh but I’d be very interested in knowing what other players/arbiters think! Thanks in advance for reading/suggestions