I accept that many (most?) people will enter the tube system at their
recharge point, but surely my situation can't be unique?

Not at all. It remains one of the irritations of the Oyster system that
unless you can nominate a tube, DLR or tram stop that you know you will be
passing through then you can't pick up tickets ordered online. I don't
really know the layout of Paddington station but you might find they have a
manned barrier (for those with luggage etc) with a validator that you can
reach and it might make you feel more comfortable to use that instead of the
automatic gates. Just a thought.

G.