Agree with the first part that says exchanges handle a lot of transactions and a daily basis but the fact is they don't use automated system for their cold storages this is because automated systems require internet connection most of the time to run and if internet connection is involved it's no longer cold storage. In my opinion I believe that most of them make use of multisignature wallet for larger amount of funds.
The automated systems don't have to handle signing keys and have access to cold storage over the internet. Like you said, that's no longer cold storage then. Automation could be used to monitor hot wallets and notify those responsible to fill them if the amount of coins drops below a certain limit. Then the refilling from cold wallets can begin offline. Maybe automated systems even start the transaction creation process for the exchange by filling out the destination addresses and amounts.