Why is there no bus service (i.e. limited-stop stage carriage) between
Leicester and Birmingham?
There must be a large demand for travel between the two, with many
companies having regional offices in Brum, 'Corps of Commissionaires'
and 'Securitas' are two that I know of, and it being a national hub
for various things.

National Express do a non-stop run between them, but not many times a
day and the first one doesn't leave Leicester until about 10 AM, while
Central Trains don't really have enough capacity, although they might
be running more after the recent resignalling.