I unironically believe if you tell all the kids they don't have to write the essay at all, much more will choose to write it.
Kids cheat not just because they're lazy. Cheating makes people feel smart. The fact you can get credits by doing very little while others work their asses off is rewarding and self-validating.
The big issue of exam-only approach is that a one-hour exam is not enough to evaluate a student's performance, unless your educational goal is just to make students memorize stuff by rote. I'd consider a 3-hour open-book exam bare minimal. But if every class does that it'll be too exhausting.