PvE does not punish keyboard turning and clicking. Movement is the most important aspect of this game and in PvE it is not required. Group PvP teaches people how to play this game better than any PvE can.
How does PvP not teach any lessons?
Movement, teamwork, priortiziation, and resource allocation are all subjects any player needs to be thinking about in PvP.
Do I stun the guy chasing me?
Who do I target?
When do I use my CC for a teammate?
How do I maximize damage while healing my team?
What’re the ranges on abilities?
Who are my teammates? Can I rely on them?
How many players need to attack target X to efficiently kill them, maximizing the damage we do while minimizing the other team does to us?
Does the target I’m chasing tend to run in certain patterns? Can I predict their movement?
It sounds to me like you’ve honestly never thought about the questions above. You can PvP like it’s PvE, sure, but that’s not a recipe for success or fun.