I don't think anything would stop them. They don't HAVE to eat deer.
Here's an idea. Take a captive cougar, radio collar it and turn it loose in one of the wild regions. See how long it survives and how many times it gets sighted. I would neuter it first though, just in case. You're right though, they don't have to eat deer. They could eat your dog or your horse or sheep or kids...they could eat anything really, just like we can. Remember Jeffrey Dahmer?