I can think of no way the battery would discharge due to a external electricl drain while having only the positive post connected. I would suspect the battery.
I wonder if it is possible that the battery, the charger, or both have the polarity reversed. I have mistakenly hooked up a battery charger backwards before and charged a battery in the reverse polarity. This is easy to do with a charger that has color coded alligator clips that come off easily, comibned with a stone dead battery. This practice could lead to a battery not holding a charge, or you could just have a bad battery, even if it is new and charged correctly. The smoked tape player may be co-incidental, but I would double check polarity, and/or suspect the battery. A battery that goes dead with only one terminal connected, must be bad or have an internal short.
