Most "beta", "gallery", or otherwise semi-private spaces I've seen on TA have on the "front door" some manner of disclaimer similar to

> Requesting access is currently a manual process — please ask in the main chat room in the 'Meta' community.

As participation grows and the number of semi-private spaces grows, it's reasonable to expect that the core TA admins might miss these requests more and more frequently. Certainly it's also reasonable to expect that not everyone will be immediately familiar with the mechanics and etiquette of The Tavern. 

Adding a formal "Request Access" setup provides a measure of certainty that your request has been seen by someone who can do something. 

I might also request that formal rejections are visible to the requestor in some way, but I can imagine this might make for a lot of moderation headache.