Well ping spams can be annoying but they might identify a situation that may require desperate attention.
It can be just a flamer pinging non-stop.
The ping-mute would not be a pleasant solution. LoL has a variety of pings that are really helpful for a team to communicate without chat. Who to Target, sinalize missing champion, asking for help, etc..
You can report if you please but people at the tribunal have no proof they were pinging like hell.
Your best bet is to politely ask them to ping less and to ping more
effectively. Because pings are extremely useful.
REGARDING THE EDIT
Ping flamers are not that common (At least I don't get them, maybe once a month in EUW). If asking them to stop and ping less doesn't work try lowering the game sound. Not completely removing the sound but lowering it.
If you are confortable with no-sound go for it but I wouldn't recommend it.