I'm working on a lengthy essay discussing this. It may not be relevant anymore by the time it's done, and might not really be appropriate for less wrong outside of replying to this specific post. But the basic summary is:
Yes, for the immediate future, a contest with a $300 prize is probably going to be more useful to you than trying to hire an artist for $300.
What you have to realize is that when you have a $300 contest with 20 participants, the actual financial payoff to the participants is essentially $15. The amount of time it takes to make a good logo is variable, depending on whether you get a flash of insight and do the whole thing in 15 minutes or take several hours trying various ideas. But essentially, on average, you are paying less than minimum wage for highly skilled labor. Doing a good job requires not only years of work, but studying design and branding principles that usually require a college education to understand and put into practice.
I don't actually know whether it's in your (or any other client's) long term interest to do this. We're dealing with a complex system here with a lot of potential long term side effects. People respond to contests irrationally, and it's entirely possible that they will continue to do so indefinitely and the payoff will always be worthwhile. But at least keep in the back of your head that what you're actually paying is a few dollars an hour, and the success you get is dependent (at best) on young hobbyists who are happy to do the work anyway, and (at worst) widespread irrationality and poor math skills among the general population.
(Note: I'm specifically talking about contest-labor. There are other forms of crowd-sourcing that I'm less concerned about.)
In the meantime, I'm also reminded of this discussion item about branding biodiversity. The philosophy behind that is definitely relevant here.
I think there must be a lot of people who would like to work in their own time over the internet. I don't know if they are schoolkids, housewifes, or workers from the third world, but - whoever they are - there seem to be a lot of them. What there aren't enough of are enough opportunities to satisfy them all. If you look at some of the "hire an expert" sites, the prices seem pretty awful there as well - e.g.: http://www.scriptlance.com/
It does appear that prizes are good motivators - in comparison with a process involving screening applicants ...
If you have graphic design experience, check out the on-going logo design competition at 99designs for the Singularity Institute. There are still 6 days left to enter and be eligible to win the $295 prize if your design is selected. Tell your friends with graphic design experience too. There are very few submissions currently.
Note: This is a blind contest. Designers can only see their own entries. All designs will be revealed when the contest ends.
If you're interested at getting a peek at the designs, they will be online after the competition is over. This is standard practice in 99designs contests to prevent designers from contaminating each other and having all the designs drift in a certain direction.