A single pulse pair is not enough of an identification. As Gerry explained in the linked post, a pseudorandom number generator varies the pulse pairs. If you vary and repeat one pulse pair, the timing difference will be easily repeatable by other units and will not be unique.
So what the unit does is generate a train of pulse pairs, in which each pair is randomly shifted. And the unit listens for the same train back, but how?
In an analog circuit, the unit will interrogate then listen at a specific time slot (range gate), in steps of e.g. 12 µs, but given the range of 2.5 ms Gerry mentioned, those are ~200 steps. To go through them quicker, the unit increases the rate of interrogating/listening. Otherwise it can be upwards of 10 seconds to get a DME reading.
In digital circuits, the replies are stored and processed (after each cycle), and to avoid errors, consecutive valid replies are needed, e.g. ten in a row, not just one, so again a faster rate helps get those valid replies more quickly.
Once locked on and the scan range is narrowed down, the unit uses the slower-rate track mode.
Also note that the average allowed is 30 PP/S (otherwise a single DME ground station can't service many airplanes), and since the search mode is faster than 30, the track mode will typically be less than 30 (that's why Gerry wrote "up to"), so in the end the average will be around 30.
Ref. and further reading: Miller, Wayne L., and Ernest J. Carnicelli. "DME apparatus and method." U.S. Patent No. 4,028,698. 7 Jun. 1977.
(The above Cessna/Sperry/Honeywell patent explains the basics before explaining their own method.)