How can some exchanges/services afford to pay for customers transaction fees?
They can avoid the fees in two ways.

First, they can choose to batch multiple small transactions together into a single transaction with multiple outputs.

Second, since they have a large pool of bitcoins, they can more easily form transactions that qualify to be free transactions, for payouts over 0.01 BTC.

The most efficient way to form these free transactions would be to make a transaction that's nearly 1000 bytes, and is sent from inputs that destroy 4 Bitcoin days (e.g. a 4-day old 1 BTC input, or 2-day old 2 BTC input). This will ensure that it meets the priority requirements.