I'm a little surprised that the tweeters can produce any sound at all. You said they barely produce sound. Usually when tweeters get fried from being over driven, perhaps by a clipped signal, the can't make any sound at all.
It is unlikely that components on the crossover boards are causing this problem. These components rarely fail, and if they do, the result would be no signal at all going to the tweeters.
It seems like you may have to replace the tweeters. Contact Sony customer service (good luck with that

) for more information. I guess you may have to decide if it is worth repairing these speakers, based on how much you payed for them, and how much Sony wants to replace both tweeters. You will be better off that way than if you search for a suitable drop-in replacement.
Those other specs, 130 watts max input power and 8 ohms impedance, refer to the speaker as a whole, but is obviously related to the individual components. Exposing the speakers to more that 130 watts for some unspecified length of time may cause them to fail. I can't say for sure without hearing those speakers, but it is likely that playing them at full volume with a 130 watt amp would be too loud to listen to for long. It is more likely that they were damaged by turning up a weaker amp up so high that it clipped.
Good luck with your repairs.