It can be a service offered by an exchange or by another company. Coinbase and Binance have OTC desks, for example.

Not exchanges. The reason is that, as GP indicated - oftentimes execution is spread across venues.

This is typically a role of a dedicated broker house or there is a special area in many investment banks that takes care of that as part of customer flow handling.

Usually OTC desks are seperate and are part of a prop trading firm.