From that we don't get how long it was between the time the Valiant left Earth, until the time it "disappeared." Couldn't have been too many years, but still maybe not immediately after departure.
...It's only later, when we learn that the episode must have been taking place in the 2260s, that it becomes necessary to assume that the Earth-> disappearance must indeed have been very
short, at most two years by modern reckoning.
If Valiant left after 2018 (according to "Space Seed" space travel stopped requiring sleeper ships after 2018) then that would be a 242 year span till 2260. Valiant could've been flying around for a few decades before being lost 200 years...
And since TOS' suggests impulse can reach FTL speeds so Valiant could've been FTL-capable without warp drive and it wouldn't contradict TNG's Cochrane's warp flight in "First Contact" in 2065. But other timeline events are different between series/continuities which could suggest WNMHGB might have happened differently (or not at all) in TNG-verse, IMHO.