Usually yes - I often make the tepache then do a second fermentation with pineapple juice (just strain the tepache back into its pitcher, mix with pineapple juice, stir daily till fizzy) because I am always disappointed with the amount I get from a gallon batch - ginger beer when I make it in a gallon pitcher makes a gallon. Tepache, so much displacement from the peels! This way it gets lightly alcoholic but not so dry.
I've noticed the same thing; the peels take up so much space. Have you tried doing an anaerobic second fermentation to really crank up the fizz and encourage yeast growth?
Yes, I do put it in a closed bottle when it's done but immediately refrigerate because it's so active, gets scary pressure if left out. It's funny - beer is fussy as heck, ginger beer only works for me with organic ginger, but tepache? Toss peels in a pitcher, add sweet cinnamon lemonade, it's almost foolproof. Pineapple yeast must be strong as heck.
From what I've read you need organic ginger as they irradiate the normal stuff, and it kills off everything. I guess you could probably get around it with a good inoculation from a prior brew. Costco was selling boxes of organic ginger a while ago, I need to check if they still have some and brew some beer.