Why did Elrond remain in Rivendell?
Elrond remained in Rivendell during the majority of the Third Age because he had established a function as a great healer.
“Elrond symbolises throughout the ancient wisdom, and his house represents lore – the preservation in reverent memory memory concerning the good, wise, and beautiful. It is not a scene of action but of reflection.”
J.R.R. Tolkien, Letters
In the Hobbit and LOTR, the characters always visited Rivendell on the way to their deeds, or adventures. Furthermore, Elrond is the keeper of the Elven Ring, Vilya, which started in the Second Age during his and Gil-galad’s forming of the Last Alliance of the Second Age.
Rivendell was viewed as an enchanted sanctuary in the “extreme eastern margin of the wesern lands.”
Thus, after “escaping” to Elrond from imminent pursuit of evil, the hero can go face it in an entirely new direction.
Did Elrond ever fight in battle?
He has fought in the Second Age and beginning of the Third Age. However, I think it’s worth noting:
Many Elven-men were great healers as they were skilled in the lore of healing bodies, though such men abstained from hunting, and went not to war until the last need.
JRR Tolkien, “Laws and Customs of the Eldar”
Did you wonder why would Elrond remain in Rivendell during most of the Third Age, or were you happy with what had happened?