@Lobakgo
OP hasn't said she's going through her home insurance. In fact she's specifically said they've already refused the claim. She's talking about going to the man's car insurance company. She also has the option to bring a damages claim against him directly.
I agree proving that the fence wouldn't have been blown down anyway is an uphill battle.
The reasons given by home insurance for refusing a claim (I'm not sure if OP actually tried to make a claim or just got initial advice from home insurers) are likely similar to those motor insurers will try to use.
This is my point. It's not straightforward. It's an uphill battle as you say. It's worth noting WHY home insurance may not cover this and the reasons they'd likely give, because it's directly relevant to how motor insurers will view the circumstances.
If you have the option of claiming through your own home insurance (and I understand OP might not), as long as the excess and increase in premiums etc make it worthwhile then I would always recommend going down this route, even when a recovery is not guaranteed.
MANY third party insurers will try and get out of paying anything they can. They'll agree the lowest contribution they think they can get away with, they'll try and deny things that would have possibly been included through home insurance (matching items etc.) - they're just generally very difficult to deal with.
As I've said, I don't know exactly what OP's home insurers told her. Whether they told her over the phone storm damage to fences isn't covered (it generally isn't) so they probably wouldn't cover damages, or whether a loss adjuster visited and repudiated the claim etc. I personally wouldn't refuse cover on a claim like this with these specific circumstances and the initial damage from impact without a loss adjuster visit and full assessment on site.
OP came here looking for advice and I think it's relevant to try and cover all bases.