No. Everything is just the same
Let say bitcoin transaction is like a banknote. You can write something on a banknote but the note itself is still valid. When gox sending a banknote to its customer, they take a picture of the note, and use the picture of the note as an evidence of delivery. Some customer, however, write something on the note when they get it from gox, and claim they have not received the note. Since the note looks different from the photo, gox can't recognize it and wrongly believes that the note is not delivered, and send another note to the customer (so the customer gets double paid by exploiting the gox's bug). Since gox believe the original said note is not spent, they try to send it to a different customer. Of course this won't work and led to all those bitcoin withdraw problem we have seen.
So gox now proposes to use a different method to track the banknote. Instead of taking a photo, they propose to use the unique serial number on every note for tracking propose.
Bitcoin is still the bitcoin we know yesterday
Following this analogy how do other exchanges tackle this problem?
Well they don't tackle it because they don't need to. Their transactions are correctly formed, and are readily accepted by the nodes and miners without modification. To force the network to accept modified transaction would take some effort now, because current version of bitcoin node would not retransmit non-canonical transaction. This is actually what made this attack on MtGox possible - and not the speedy link to the miners, or significant mining power of the exploiters. And that's another implied lie in their statement. MtGox issued not-quite-correct transactions to start with, they were rejected by the nodes, and then replayed by the hackers with fixed format. Now I hope you get a better picture of how filthy their lies are.
UPDATE: In the event there are indeed any rejected transactions, they are very rare and far apart, can be easily investigated and dealt with appropriately.