How does exchanges, such as coinbase handle off-chain transaction?
Yes, exchanges tend to have huge cold wallet where they amass and send most of their bitcoins ( you can even follow those addresses by each exchange on the top 100 richest bitcoin addresses )

When a user want to withdraw fund they can use smaller hot wallet to send coins, some exchanges will use payment channel for the transfer but most have not adopted it, or choose not to.

For more detail about how they do that you can check this question How do cryptocurrency exchanges handle users&#39; accounts?