In my experience, there is no lack of demand for "artisan" venison with provenance, although some people do need a bit of educating on how to cook it.
But you do need to put yourself in front of the customers and promote your product, which is fine if you're just a small-scale "Fred in a shed" type, with plenty of time and not too many deer.
The only way that demand is likely to increase at a more mainstream level is to get venison into catering.
Either way, the quality has to be good, and unfortunately a lot isn't.
Incidentally, I attended a Farmer's Market recently that was a new venue for me, so none of the customers knew me or expected venison to be available. By the time I'd got my stall set up there were people queuing to relieve me of my product and hand me their cash.
I had only butchered two 30kg (larder weight) prickets, and within 3 hours I'd sold the whole lot (just over £650), and now I'm getting emails from customers wanting venison again at next month's market.
I honestly wish I had access to the sort of numbers of deer that some of you chaps have, because selling it really isn’t difficult.