How about taking a harder line then? Make your warning at the start of the semester, and to everybody: any reasonable suspicion of cheating results in an immediate referral to the honor board, no protesting or second chances. The first assignment rolls around, a few people are caught cheating, and you spend X hours submitting their cases to the honor board. Let the honor board handle the adjucation of the matter, which is where it belongs, and wastes less of your time. It also lets the department know that they have a problem, since some of them are probably involved with the honor board.

Students hear that peers are getting sent straight to the honor board, and they fall in line. The next assignment that rolls around, almost surely less students will attempt to cheat. If you have to spend Y hours sending more students to the honor board, hopefully Y is less than X. But it sends a loud-and-clear, unmistakable message to the students that cheating is unacceptable. With any luck the students that actually take the class seriously will appreciate your candor on the matter.

I agree that allowing students to actually come in and argue their case to you is a waste of your time. You're right, that's not your job. Send it to the honor board and waste the time of the deans and the department heads, since that's their job. If you let students waste your time by giving them an extra strike before they're out, you can't be surprised when they try to take advantage of it. One strike and you're out tells students that it doesn't matter if ex post facto you think your best friend's grandma's death drove you to do it: cheating is never acceptable.