What is the use of a synchronous callback function in JavaScript?

Some very common examples of synchroneous callbacks are the methods on Array.prototype: forEach, map, filter, etc.

The role of the callback is to provide a partial implementation that can be easily swapped in a larger algorithm. Some design patterns like template method and strategy come to mind.

You're right.
There is usually no reason to have a sync callback.

Exceptions include callbacks that might sometimes be async, or callbacks that can be raised more than once (eg, [].map()).