What's TWC ?
The Weather Channel (Click on the link I provided above for more on TWC & Dr. Lyons)

They still need the windspeed-analysis, but only as a tool to predict damage.
Yes, but the "analysis" as it is currently conveyed (by way of the Saffir-Simpson Scale) gives a horrendously unclear picture of what that damage is going to be like. (Ignores things such as the total duration of damaging winds, the radius of damaging winds, radius of exceptionally damaging winds, etc.)

BTW. why limit to 5 categories representing the windspeeds ?
Just give the speed. Or if folks can't remember their relations and significance,
extent it to fractions like Cat2.7 , see earthquakes.
Rating tropical cyclone windspeeds is not at all as simple as the Richter Scale. A "Cat 2.7" would never work, whatsoever. It would only be a worsening of an already flawed system.

I'd like several forecasters giving subjective expected damage predictions,
(expectation value,deviation) and an analysis who gives best predictions
as with the "competition" here with the several forecasters, forecasting-models.
Good luck with that! Hurricane forecasting is simply not that precise, and likely won't be for decades to come. Forecasters might have contests that could be tracked, but you're likely to get more snake oil than not, until forecasting improves markedly.

And prediction-markets trading damage-contracts, with the opportunity to
"insure" by shorting damage-contracts. Well, there are catastrophe-bonds,
but only for big institutions
I believe what you seek for the rest of us to be able to trade here already exists. (Oil futures, grain futures, etc.) While not "damage contracts," these are pretty darn close, when you think about it. As close as it's going to be given the state of the science -- although, I think more exactly of what you want might be a nifty item on Intrade.