Why is the sound quality of onboard public address so bad?
For the regulatory portion of this question.


  14 CFR 121.318
  
  No person may operate an airplane with a seating capacity of more than 19 passengers unless it is equipped with a public address system which— 
  
  (a) Is capable of operation independent of the crewmember interphone system required by§ 121.319, except for handsets, headsets, microphones, selector switches, and signaling devices; 
  
  (b) Is approved in accordance with§ 21.305 of this chapter; 
  
  (c) Is accessible for immediate use from each of two flight crewmember stations in the pilot compartment; 
  
  (d) For each required floor-level passenger emergency exit which has an adjacent flight attendant seat, has a microphone which is readily accessible to the seated flight attendant, except that one microphone may serve more than one exit, provided the proximity of the exits allows unassisted verbal communication between seated flight attendants;
  
  (e) Is capable of operation within 10 seconds by a flight attendant at each of those stations in the passenger compartment from which its use is accessible; 
  
  (f) Is audible at all passenger seats, lavatories, and flight attendant seats and work stations; and
  
  (g) For transport category airplanes manufactured on or after November 27, 1990, meets the requirements of§ 25.1423 of this chapter. 
  
  [Doc. No. 24995, 54 FR 43926, Oct. 27, 1989]
  
  14 CFR 25.1423
  
  A public address system required by this chapter must— 
  
  (a) Be powerable when the aircraft is in flight or stopped on the ground, after the shutdown or failure of all engines and auxiliary power units, or the disconnection or failure of all power sources dependent on their continued operation, for— 
  
  (1) A time duration of at least 10 minutes, including an aggregate time duration of at least 5 minutes of announcements made by flight and cabin crewmembers, considering all other loads which may remain powered by the same source when all other power sources are inoperative; and 
  
  (2) An additional time duration in its standby state appropriate or required for any other loads that are powered by the same source and that are essential to safety of flight or required during emergency conditions. 
  
  (b) Be capable of operation within 3 seconds from the time a microphone is removed from its stowage. 
  
  (c) Be intelligible at all passenger seats, lavatories, and flight attendant seats and work stations. 
  
  (d) Be designed so that no unused, unstowed microphone will render the system inoperative. 
  
  (e) Be capable of functioning independently of any required crewmember interphone system. 
  
  (f) Be accessible for immediate use from each of two flight crewmember stations in the pilot compartment. 
  
  (g) For each required floor-level passenger emergency exit which has an adjacent flight attendant seat, have a microphone which is readily accessible to the seated flight attendant, except that one microphone may serve more than one exit, provided the proximity of the exits allows unassisted verbal communication between seated flight attendants.
  
  [Doc. No. 26003, 58 FR 45229, Aug. 26, 1993, as amended by Amdt.
  25-115, 69 FR 40527, July 2, 2004]


There are systems in place to increase volume in certain instances, eg, on the 767 both engines running gives a 6 dB increase and O2 masks dropped gives an additional 3 dB gain in volume.