Barring additional detail from the forthcoming novelization I don't think there will ever be an in-universe explanation. However, the TLJ Visual Dictionary and Incredible Cross-Sections have some info about hyperspace tracking, though it seems a bit contradictory to what we see in the film.
From Incredible Cross-Sections, in the Supremacy section (snokes flying wing ship)
A Hunter's Secrets:
The First Order tracks targets throught hyperspace using a combination of technological advances and brute-force data crunching. The shipboard tracking control complex boasts the data-sifting power of a planetary intel hub, linking huge computer arrays to databanks loaded with centuries of combat reports and astrogation data.A static hyperspace field generated around the machines then accelerates their processing power to unheard of levels. A target's last known trajectory yields trillions of potential destinations, but the system can assess them with terrifying speed.
From the Visual Dictionary, again under the section on the Supremacy:
At the heart of the Supremacy's advanced hyperspace tracker is a complex static hyperspace field generator. This envelops arrays of databanks and computers in a localized hyperspace field that accelerates their calculation speeds to unimaginable rates.
So this bit of data suggests that the tracking methodology is basically predictive, based on some readings made of the ships jump into lightspeed. It heavily implies that the capability to do this rests solely in the Supremacy, not in regular FO Star Destroyers. But perhaps the trajectory data could come from another ship, but must be fed into the Supremacy supercomputers, which is why Finn and Rose try to sabotage that specific component.
Rose mentions that "hyperspace tracking is new tech", but must work like active tracking. While the principles of how it works could be generally understood even if it is new, Rose and Finn may be making an incorrect assumption but get to the correct answer anyway (which is to hit snokes ship).
By comparing it to active tracking, we can make some assumptions based on how active tracking works in our world. Active tracking implies that there is a signal emitter and a receiver (much like RADAR or SONAR for us), contrasted to passive tracking where there is just a sensor receiving some sort of emission or trace left by the target (like passive hydrophones in subs that listen for noise made by another ship). The Visual Dictionary and Cross-section books don't really support hyperspace tracking working in this way, other than perhaps to get the initial data that is then fed to the supercomputers.
So there could be several possibilities for why just the lead ship is using hyperspace tracking:
A. protocol, perhaps to prevent confusion amongst the ships about which way to go because all the pursuing ships are slaved to the lead ship in order to jump in unison. Given the distances involved in making jumps that are lightyears apart, precision and coordination is paramount. So the front ship does the tracking, gets the direction to jump, and the entire fleet uses that to jump together so that they arrive together, not scattered and liable to be destroyed piecemeal if they are lead into a trap.
B. a technological limitation, i.e. the sensor can only operate on one ship at a time to prevent masking the hyperspace trace or drowning out the sensor (kinda like if a bunch of subs start emitting sonar pings at once). It makes sense that the front ship will do the tracking so that there are no intervening ships to block it, or other emitters to confuse the return. Star Wars ship fly in VERY close formation, even compared to modern naval ships or even aircraft, so they can actually block each other pretty easily.
C. By "lead ship" Finn and Rose mean the command ship, the one leading the formation. It seems pedantic, but they talk about infiltrating Snokes big flying wing ship even though then they are leaving, Finn has a fleet map up that shows Snoke in the rear of the formation, all the other star destroyers are in front. Eventually Snoke does move his ship into the lead as seen by the effects of Holdos ramming tactic, but initially it looks to be in the back of the formation, but since Snoke is on it, it is the command vessel and could be considered the "lead ship".
I now favor A the most, the emphasis on active tracking is misleading, the hyperspace tracking just needs a trajectory. So it makes sense that the front ship with the most unobstructed view of the target is the one that monitors it and reports back the data since it needs a visual triangulation based on the lightspeed "zip" line a jumping ship leaves or something like that.
Its possible that the source books are incorrect about how many ships contain hyperspace trackers, or it was changed later on. The Supremacy wasn't shown in the initial retreat from the Resistance base, so either other destroyers can track or it is true that just one ship can make the observation and feed it to the Supremacy wherever it happens to be. I think Rose and Finn make some incorrect assumptions about the tech but chanced upon a possibly successful plan anyway, given Finn's knowledge of the Supremacy.