So frustrating....

[user 43495]

You haven't said why your brews had to be poured away so it's hard to guess. I have used king kegs for years now; I used to employ collapsible polypins because in those days you could easily get them for free and as they collasped when the beer was drawn off they didn't require gas nor did they let any air it. A KK keeps the beer better for longer, so I have found. One had about a gallon left in it before I went to work away for a month and upon my return the beer was still excellent. I have never used the top tap versions though. I have a stillage in my cellar. Now, assuming you poured the beers away because of sourness that would suggest air had got in or the sterilisation was insufficient. Re beer reluctantly pouring out of the tap, I would punt your fermentation hadn't produced sufficient dextrins to allow for gas to be produced in the barrel. One trick to rescue a flat beer is to pour in a pint or two of vigorously fermenting wort from a new brew into the flat one and leave it for a while. Otherwise dissolve a few ounces of ordinary sugar in some hot water and throw that in. Then leave it for a few days. It's better if you can move the flat barrel to a warmer place than that used to stillage to revive the fermentation.