If you’re going to use “driven” in this context, I think you’ll need to go with “by” as your proposition, followed by the person or (more often) the concern defining the conversation or moving it forward (not, as in your example, the destination of the driving).
Example: The conversation was driven by Andrew's love of all things British.
or: The conversation was driven by their shared desire not to harm the bees.
To stay closer to the statement you’re looking for, you rightly suspect that “steering” will work perfectly.
As for the preposition, I recommend “toward(s)” as in:
I managed to steer the conversation towards school bus fires so that I could tell my heroic tale.