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The GLI isn't for his website per se, instead, its for his company...
Its not like car insurance where they'll eventually replace your car (after a long drawn out legal battle) or give you a payment of some sorts. The server/site isn't insured where they'll say "oh, server's gone - $1384!" Instead, liability insurance protects the assets of a business when it is sued for something it did (or didn't do) to cause an injury or property damage.
In this case, if you're sued because of something they cannot go after your assets (thanks to your LLC status) or that of your company because the insurer is paying for your legal costs. Your lawyer should have explained this better; now I think you're somewhere way off base I'm afraid.