Zefram Cochrane discovered warp-drive 200 years before Kirk's 5-year mission. The SS Valient has been missing for the same 200 years. It's not sensible that the first F-T-L Earth ships would be sent on a non-stop voyage to the edge of the galaxy, ignoring the exploration of everything between Earth and the galaxy edge. At a continous warp-5 speed (100 times light speed), it would require 150 years (non-stop) travel time to arrive there. Spock describes the SS Valient as an Earth ship, not a Federation ship. Did pre-federation ships possess warp-5 speed? And were they 'sleeper-ships' to accomidate the 150-year travel time at warp-5?
On the point of the edge of the galaxy being the top or bottom edge... the barrior is the same location the Enterprise had to pass on it's way to Andromeda galaxy in: "By Any Other Name" episode. Which side of our galaxy faces the Andromeda galaxy?
This argument rests on the assumption that warp velocities are simply the warp factor cubed times the speed of light and that's it. If that were the case, it seems unlikely that Kirk's Enterprise would be able to make it there and back, plus have any other kind of time to explore anything else.
I'm glad "Star Trek Maps" (Bantam, 1980) came along and added a modifier (Cochrane's Variable, which changes depending on environment) to boost warp velocities in certain parts of the Galaxy (I prefer to think that dark matter and dark energy play a role, especially given how the Enterprise broke down upon encountering the Barrier) while leaving other areas out of reach. This also has the side-benefit of explaining why starships in any generation of TREK seem to spend so much time at Warp 1-to-2 and yet they don't take forever to get anywhere.
There's also another advantage to Cochrane's variable here: maybe the old Valiant tried to ride a wave of dark matter or energy in deep space to get a "tail wind" and the resulting storm lead to the ship's demise at the Galaxy's edge.