Someone does operate it. The first of the daily three Orleans express buses that run into Rimouski from Montreal arrives in Rimouski around 2:30 pm. Passengers can then connect to 2 buses from Rimouski to the town of Gaspé. One of these connecting buses follows the north road, and the other follows the south road, essentially splitting the first Montreal-Rimouski bus. The other 2 buses from Montreal to Rimouski arrive later in the day, without connecting buses.
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Honestly, the biggest argument against better train service either to Rimouski or to New Carlisle is that these regions are already served by buses. But if the Gaspé region is so determined to have train service 3 days per week to New Carlisle, with a connecting bus to Gaspé town, I would argue both VIA and the Gaspé region might be better off establishing daily service to Rimouski instead.