Speaking from my experience as a moderator on other sites that have had (or proposed) contests: the contests that get the most support from Stack Exchange (e.g. for prizes) are the ones that improve the site in ways that help the site's mission. On Writers we had a contest a couple months ago, tying into National Novel-Writing Month, to build quality content around genre tags. This got us some great new questions and new users who've stuck around -- just what we needed.
Contrariwise, contests that are orthogonal to the site are not likely to get traction. Would "how to harmonize this melody" questions be good questions here if there weren't a contest tie-in? I don't think so; they're subjective, opinion-based, and not likely to be of broad, long-term interest (they don't help future readers). So before pursing a contest, it's best to figure out what the site needs and build a proposal around that.
Now, none of that stops any individual (or group of individuals) from offering to sponsor a contest, as one of the top users did on Board & Card Games about a year ago. He offered his own prizes and set his own rules. The contest benefitted the site (rewarded the things the site needed) so it got good participation. (Possibly he could have asked for SE sponsorship for it, but he didn't.)
So you are free to offer your own contest based on anything you like -- but even if it doesn't help the site, you do still want to make sure it won't harm the site. A contest that calls for types of questions that will just get closed will only frustrate people.
I don't think this specific contest proposal aligns well with the site, but if you want to promote the site I encourage you to think about what the site needs and what kinds of contests could help with that. And if you're willing to sponsor the contest yourself, that removes one hurdle.
(In my experience prizes don't need to be major to motivate people. On Mi Yodeya a couple years ago we had a contest for a ~$20 gift card; the point was the contest and the bragging rights, not the cash value of the prize. The BC&G contest had prizes of (used) individual board games.)