You did miss a great deal. You also have the timing of the sub's visits to San Francisco and San Diego reversed.
The sub goes to San Francisco first, after coming down from the north coast of Alaska near Point Barrow. What happens there is that the ship's Yeoman, Ralph Swaim, gets out through the escape hatch and swims ashore because SF is his hometown and he wishes to die there. Before leaving the next morning the sub sees Swaim in a boat fishing and surfaces. He and the Captain talk for a couple of minutes (Swaim tells what he's found on shore, then apologizes for jumping ship but explains his reasons, and the Captain says he understands), before the sub leaves for good.
They do not find the Coke bottle on the radio key in San Francisco. It's in a radio transmission station at an oil refinery near San Diego, where they had previously pinpointed it. They go there after visiting San Francisco.
As to the power source, there were a couple of discussions about it before they went there, but the officer put ashore at the refinery found it when he walked by a powerhouse whose equipment was still running. (Just before this he had seen a partially damaged neon sign still lit, so he knew there was power.) After he locates the Coke bottle tugging on the window shade and striking the transmission key underneath, the officer sends a clear signal telling what he's found and, on his way back to the sub, stops at the powerhouse to shut off the turbines so there will be no more electrical power.
So, no goofs anywhere, you just watched too much football!
Incidentally, in the book this whole scenario is handled differently. The sub voyages north along the U.S. Pacific coast. Originally they were supposed to look in on the Panama Canal and San Diego but these were struck off their list due to uncertainties about minefields. They only look at San Francisco from five miles outside the Bay, observing that the Golden Gate Bridge is largely destroyed and that all the buildings have been burned and blasted. They then go to Puget Sound, where Yeoman Swaim jumps ship at his hometown of Edmonds, Washington (not SF). The radio transmissions have been identified as coming from the vicinity of Seattle (which is undamaged), not San Diego, and when they get close they specifically locate it at a U.S. Navy base on Santa Maria Island. They put an officer ashore and he finds, not a Coke bottle tugging on a window shade, but that a casement window has blown in and that, as it teeters in the wind half in and half outside the radio room, it strikes the live radio key just beneath it. The officer replaces the window, transmits what he's found, and closes down the set and station. On their way back through the Sound they have the encounter with Swaim in his boat. We learn later that the sub went on, not to the Arctic Ocean, but to the Gulf of Alaska in the south, to check on the "Jorgensen effect" (which, as in the movie, does not exist), then to Pearl Harbor (which is destroyed), before returning to Australia. (Also, in the book the entire Southern Hemisphere survived the atomic war, not just Australia, a plot point in the movie which makes absolutely no sense.)
Also, they learned on their trip up the coast that there was still hydroelectric power by observing lights on shore throughout the state of Washington, including in Seattle. The powerhouse on Santa Maria Island is still running and the officer, Sundstrom, shuts it off before leaving because he couldn't bear the thought of leaving it until it cracked up (there was a grating noise beneath the hum of the machinery after two years without maintenance and he felt it couldn't last much longer).
The book is better and much more logical.
By the way, one reason they switched the location of the radio transmission from a U.S. Navy Base in the book to a private oil refinery in the movie is the the U.S. military refused any cooperation with the filmmakers, on the grounds that they objected to the idea that a nuclear war would cause such massive and fatal destruction. Hence, no naval base, and they refused to loan out a nuclear submarine for use in the movie; producer-director Stanley Kramer had to use a diesel-powered Australian sub for the film. (They also changed the name of the sub from Scorpion in the book to Sawfish in the film; there later was a U.S.S. Scorpion that sank off the Azores in 1968.) And for those opening shots on the Golden Gate Bridge in San Francisco, Kramer was given 90 seconds to film them on the roadway before they had to allow the traffic back on!
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