Oddly enough for a big fan of Run Silent, Run Deep, the novel, I haven’t seen the movie. However, the novel was written by Capt. Edward L. Beach, a distinguished submariner in WWII and the post-war era and is generally recognized to be extremely accurate.
Most of the incidents in the novel (and the movie, I presume) were based on real life WWII incidents, and many of those were based on exploits of the U.S.S. Wahoo, commanded by Dudley “Mush” Morton and the U.S.S. Tang, commanded by Richard H. O’Kane, two of the war’s most acomplished submarines. Admiral O’Kane, who served as the Executive Officer of the Wahoo before commanding the Tang, wrote two memoirs of his service on those ships, Wahoo: The Patrols of America’s Most Famous World War II Submarine, and Clear the Bridge!: The War Patrols of the U.S.S. Tang. In Wahoo, O’Kane describes a “down the throat” shot against a Japanese destroyer.
As to attacking the merchant ships before the warships, that was simply sensible strategy for the stealthy but relatively slow WWII submarine. First, attacking a small, fast, manueverable destroyer or other escort ship was more difficult than attacking a slow merchantman. And if the merchant ship attack could happen unobserved or away from escorts, the submarine might be able to attack multiple ships in the convoy before being forced off. This would be less likely to happen against the watchful naval crew of an escort ship.
Second, destroying a merchantman (particularly a tanker) would deprive the Japanese of both the ship and its cargo, while sinking a destroyer would likely cause little damage beyond loss of the ship (many of the crew would likely be rescued in either case).
Third, convoys were often guarded by multiple escorts, so sinking one would bring the others out against the submarine, forcing it to break off its pursuit of the merchant ships. And even if all escorts could be destroyed, by the time they were dealt with, the merchantman sould probably escape by running away, as the submarine had little speed advantage that it could use to chase down a fleeing ship.
Of course, if a a submarine could target a major capital ship like a battleship or a carrier it would go after it, but attacking destroyers and other small escorts just wasn’t worth the high risk and low reward for the U.S. WWII submarine fleet, which was primarily engaged in commerce rading.