They also remember the writers they loved and the agent didn't. Agencies have different methods of handling the mail and scripts. In some agencies, the standard practice is to open the mail daily. I've heard about one agency where the agent, subagents and the assistants collect all the letters once a month and sit around a table reading them and as a group decide which scripts to ask for and which to reject. Also, I have not heard of an agency where answering the mail is anyone's only duty.
Query letters are silent slips of paper that can simply be dealt with later when there is time.
Contrary to how you may think, they don't enjoy rejecting people, so it's not a task they leap to do. Because it means that someone whose judgement they trust has read your script, may know a bit about you, and feels that reading it is not going to be a waste of the agent's time. It also means you know enough about the industry to get the script to someone with some clout. Small contests or ones with a bad reputation do not impress an agent. The contests out of UCLA and USC are highly regarded as well.