"whose main function was to make noise when submerged so enemy escorts could find them easier."
Until the GUPPY class, none of the US subs could go fast enough for long enough submerged to make either streamlining or flow noise a real issue. I understand your distaste for deck guns and AA guns, but the theory at the time was if caught in shallow water and/or with low batteries, it was better to have secondary weapons and not need them, than to need them and not have them.
"the relative bearing in the periscope IS part of the equation for determining the AOB"
Nope. If your heading is 000 true and the relative bearing is 090 he's off your starboard beam and also at 090 true. If he's heading 270 the AOB is 0, if he's heading 180 the AOB is Starboard 90, if he's heading 315 the AOB is Port 45. The Angle On the Bow is how YOU bear from HIM, not his bearing from you.
As for the PK, the primary purpose for that was to keep track of his "probable" location while the periscope was down - in the game you can unrealistically leave the scope sticking up indefinitely outside 1000 yards, in real life they tried to keep the exposure down to a 10 second observation every 2-3 minutes, which made the PK actually useful. Again the PK assumes no change of speed or course by the target between observations, it was considered a "working fiction" strictly to assist the tracking.