It's not only about trusting the software itself (BitcoinABC) but about trusting the exchange.
Think about it: You would be sending your entire Bitcoin savings into a single point of failure, which is linked to an exchange, which may or not have your personal data, including IP, real name, real address, picture of your ID if you want bigger withdraw limit...
Errrr why are you sending all your BTC to an exchange?
![Huh](https://bitcointalk.org/Smileys/default/huh.gif)
The new BTC wallet can be
any BTC wallet (ie. a software wallet on your PC, a hardware wallet, a paper wallet or an exchange etc)... where you send it is up to you. The point is to put your BTC on a different seed/keys, so that should the BCC wallet you choose to use be "Bad"™ and steal your seed/keys... the thief will only be able to access your BCC, as your BTC is already some place else.
It doesn't matter if its BTC or BCC, the point is, you would be sending your entire Bitcoin savings (in whichever blockchain) and the exchange would know, because generally exchanges don't create more than 1 deposit address by account, which means, you would be telling them (I got all of these coins, all of these addresses are mine).
Also with some simple blockchain inspection, they can see that you moved your coins from wallet A to wallet B after the split, so they can also find your BTC savings.
Bitcoin's lack of fungibility: an huge problem.