In message , Tim Roll-Pickering
writes

That explains why it was built but not why the connection at Holborn is so
useless.

I suspect that by the time it was built, it was already obvious that
Aldwych would never be more than a rather useless stub. During the
planning stages it must have seemed to have had great potential to
extend to Waterloo - but an attempt to get as far as Temple was killed
off by the LCC and local landowners in 1902. Then a bill for a single
bore tunnel direct to Waterloo was killed off by parliament in 1905 - by
which time work had already begun on the Aldwych branch.

So I suspect that the odd crossover arrangement at Holborn was
eventually never seen as anything much more than a way of getting the
Aldwych shuttle car out onto the main line for repairs, etc. (despite
the very short-lived theatre through train).

--
Paul Terry