In order for that explanation to make sense, one has to assume that 11x multipliers and 25x dividers were common off-the-shelf items during the time period in question.

If the choice is between a custom crystal and two custom boards, the custom crystal is going to be the far less painful choice.

I would bet that the BBC was using the 1 MHz crystal and the associated division/multiplication circuitry for other frequency control purposes, not just the musical test tone. 1 MHz is smack-dab in the middle of the medium wave band usually used for AM broadcasting, and the BBC has always broadcast on many different frequencies at the same time.

Having all the broadcast frequencies controlled by a single reference crystal would have some obvious advantages (and a few disadvantages too, of course).

Arbitrary dividers and multipliers are fairly easy to design in a circuit. Certainly far easier than cutting a custom crystal.