Why does a DME ground station wait 50 microseconds?
The delay is not 50 milliseconds, but 50 microseconds.

It takes some time for the ground equipment to decode the incoming signal, decide it needs to reply, and fire up its transmitter. In order for the distance measurement to be reliable, this inevitable delay needs to be precisely defined and the same for all DME stations. It also gives the airborne interrogation some minimal time to shut down its transmitter and start listening even when it's right on top of the ground station.

50 µs must have felt like a suitable interval to standardize on, long enough for it to be reasonably possible to engineer the ground stations to meet it (given the technology of the time the design was made), yet short enough that one DME station can service an acceptable number of aircraft without needing to handle overlapping delays between different interrogations.

The standard delay also means that a DME station used for ILS can be located midway between the runway's thresholds and adjusted such that aircraft see the distance to the threshold rather than the distance to the physical antenna, by reducing the delay appropriately. (Such a DME does not give useful readings for aircraft not aligned with the runway).

Note, by the way, that the 50 µs is the time between receiving the second pulse of an interrogation pulse-pair and transmitting the second pulse of the resply. So the ground station actually has to start transmitting the reply less than 50 µs after it knows it has received a complete interrogation. For Y channels where there are 30 µs between the two reply pulses, the ground station actually has only 20 µs to react.