Make cycle awareness a core part of the driving test

Teach people how to drive around cyclists and at least some of them will remember it; surely that's better than not being taught at all?

Yes, you can try and teach them. Just as you can try and teach them not to exceed speed limit, not to drive too close behind each other, not to pull out in front of cars etc, etc,.
The drivers that listen and continue to drive as they were taught are in the minority and they are the ones that already treat cyclists as they should be.

You need to get to the impatient, poor drivers who don't give a shit about cyclists and see them as an obstacle to their progress and must pass as quickly as possible at all costs (blind bends, oncoming cars and so on). The test does nothing to stop that.