It turns out that's not really true. The calculations are pretty pointless now. Except as a really expensive random number generator.
[EDIT]
I wrote on this last night.
https://bitcointalksearch.org/topic/m.1826040
The textbook attack is the double-spend - spend the same coin with two merchants, and make off with their goods before they can realize what happened. To prevent this, everyone needs to know what coins have been spent or not spent.
Who do we trust to say which transaction actually happened first? Your proposal seems to be, okay, we trust the exchanges. If they disagree, they all die, so they'll surely work out their differences and agree on the transaction order.
In the meantime, can you trick Alice into giving you something for coins that have already gone to Betty?
If a "trustworthy exchange" has secretly become insolvent for whatever reason, can't it generate a massive golden parachute for itself by assisting an evildoer in making a big double-spend? Even if it fails afterwards - it was going to fail anyway!
If those exchanges get shut down for whatever reason, what then?
The whole point of Bitcoin over earlier cryptocurrency experiments is to stop having to rely on any given party (e.g. the exchanges) to be a trusted bookkeeper and dictate the "true" order of transactions. Instead you trust that "all trustworthy people" can outvote "any given villain" in computational power.
If you think the former is sufficient, there's nothing wrong with that. But Bitcoin-type cryptocurrencies (and indeed, blockchains generally) are trying to solve a problem you don't have.
What do you look for in a cryptocurrency, if I may ask? What features are essential to you?