By opening a contest, you're not only going to get a much lower quality of submissions, but in the end, the final decision is going to be made by you, someone "useless at anything artistic".
Paying a designer $500 won't get you a full brand identity makeover, but that's not what you're going for. You should be able to find a talented designer who will spend an hour discussing what you want to get our of your logo, some time going over a number of their own ideas, and then wrap up the project and give you the sizes you need and a vector file if you ever need to scale your logo up or down.