The second point you brought up is false. After the split it will not be possible to spend your coins twice on the same chain. The chains will have effectively split creating two separate currencies, it does not make double spends possible, for you to say that is just baseless fear mongering.
He doesn't mean spend the same coins twice on the same chain, he means that with two chains, the amount of Bitcoin you have essentially doubles. If you mix your prefork coins with post fork coins on both chains, then you have the same amount of currency on both chains. You have double of what you had originally and can thus spend more than what you should have been able to.
This is a good thing, it protects the users of Bitcoin, since regardless of what chain becomes dominant Bitcoin holders will have a share in both chains, making it a win win for users. Surely it would not be better if users had to choose? Making it a win lose proposition. This scenario would effectively split Bitcoin into two currencies, therefore it does not increase the total supply and does not weaken the value proposition of Bitcoin, from my perspective it actually strengthens it.
But for the miners mining on one chain, if that were to lose, they would lose a lot of money. For businesses, which coin should they accept? Some people might take advantage of both and accept coins from both chains. Then people will have twice the money that they should have, essentially inflation. Other businesses might not know what to do and simply stop accepting Bitcoin until there is only one coin that is called Bitcoin.
Which coin is the "real" Bitcoin? You can't have two coins calling themselves Bitcoin.
Then there is another technical aspect, since both would still share the same magic bytes, the same port numbers, and some other stuff that I don't remember right now. That means that blocks and transactions for one chain will be broadcasted to nodes that support the other chain. This also means that it is possible for a node to be accidentally isolated from the network if all of its peers happen to be for the chain other than the one it uses. Then it wouldn't receive any blocks or transactions and any transaction it makes wouldn't be seen by the network. This can cause problems for both users and miners. Miners would have a higher orphan rate on both chains because their blocks would propagate slower as some nodes would relay them and some wouldn't. Same with transactions. Users would have a harder time with getting their transactions out. Both could simply become unsynced with the network. Having those two chains could probably cause some more problems that we haven't even thought of yet.