If you had the "old" coins, would you want to spend them in the "new" network where their value is much lower?
The coins are copied. You could spend them in both networks independently.
Ah, so you are saying that I can use the same coin twice: once on each network. Basically, as long as the networks keep splitting the coins can be re-used. However, this would devalue the coin, independently of whether the new networks introduce further inflation. This seems undesirable to most people on all the networks: if by splitting I can still only buy 10 cups of coffee with my bitcoins, just now it's more of a pain because I have to remember through which network to use it, I wouldn't want to do it.
If someone started a new Bitcoin-like currency they should start from a blank slate. Cloning the network chains would be confusing and frivolous at best.