If it is an Ethereum transaction, why does a transfer to a crypto exchange mean any danger?? I do not get it at all! It was supposed to be as dangerous (or not dangerous) as any other wallet address.
They are just worried about an exchange to collapse maybe that's the point of that reminder, however it shouldn't be told that way.
As long as you are going to use a legit and trusted exchanger there's no problem at all. That's just a friendly reminder from them.
It's up to you if you will follow them.
Iย think you are wrong with your summation, the private keys in essential because the Chronotokens on exchange and Ether on exchange will fall on different addresses. though if the exchange has both Ether and Chrono tokens traded on it, this might be rectified if you contact them