Significant orbital play on the output side seems to be normal for tapping heads. I've yet to see one of any breed that is even approaching rigid. Certainly my full set of Pollard heads are all somewhat wobbly.
I'd always assumed that they were designed that way to cope with any small misalignment between tap thread and the hole to be tapped. They are, after all, fundamentally drilling machine tools so rigid mounting accuracy to a thou or less may be hard to achieve. Especially as common practice was to set-up the tapping head on a different machine to that used for drilling.
Rigid mis-alignment will likely snap small taps and create malformed threads with large ones. I found out the hard way that the combination of Pollard no 3 tapping head and Pollard 15 AY drill was quite capable of snapping a 10 mm tap when carelessly driven into the base of a blind hole. Gear drive is effective, if a bit noisy, but it doesn't slip like a belt when overloaded. Oops!
Clive