Thank you for your a question.
History has many narrations on Abal fadhl al-Abbas (as) and also if he fought on the day of Ashura or not.
Some strong narrations mention that he did fight and killed many.
However, the instance when he was killed, he had only gone to fetch some water and it was not on the battle field rather through the palms toward the Euphrates.
The enemies ambushed him while he had not gone for a battle but just water.
Also, Imam ali asked for a brave son, and Hz.Abbas is actually the exact thing that he (Imam Ali ) asked for.. Do we ever think how much courage it needs for someone to put down a task, that he is best at, by his Imam of the time!? And get killed for the same!? (The real bravery is to be obedient towards Wilayat I.e the Imam).
Or how much courage it needs not to drink water after holding it in your hand (and being super thirsty) for the love of your Imam!?
Al -Abbas is al-Abbas because of what he did on the day of Ashura. Otherwise he would have fallen from his lofty position of Bab al-Hawaij, had he been killed like a martyr during battle or won it. ( the Martyrs of kerbala are of course on a totally different level than others, however al-Abbas outweighs all the other sacrifices and shines)
I hope this answers your question