Think the headstock tuner argument is bogus, since they work from vibrations. I've used them before with no issues.
Inline tuners work fine provided you have an amplified banjo. If it's purely acoustic you can't mute the natural sound. If it uses tuning pegs and gut strings, I'm surprised that it's too loud, that sort of banjo idoesn't normally make as much volume as something modern with steel strings and a tone ring.
If it is too loud, a DIY mute in the form of a scarf or item of clothing wedged between the rod and the vellum will quieten it down a bit. Bonus style points may be awarded for using a soft toy instead, I have a Gabumon from Digimon living in one of my banjos for exactly this purpose.
Tuning issues are sadly quite common with friction pegs. I eventually gave in and replaced the pegs on my Windsor Premier with some geared banjo tuners, and it now holds it's tuning very much better. Even the more budget models with geared guitar-type machines are better in this respect.
There's not much you can do about someone who insists on tuning up over song introductions. Usually this is less the fault of the banjo, and more the banjo player being an inconsiderate derrière. If he's deaf you might need to point out how much it is affecting the audience.