Do you want to deflect customers? Send people elsewhere? Stop them from contacting you?
Or do you want to bring them closer? Collaborate with them? And understand their wants and needs better?
How about instead of measuring tickets deflected, you measure insights gains?
How many insights have been implemented into the product/service? How many of them have changed the service or your company in some way?
When this becomes an agreed goal, your actions change. You focus less on speed of response and driving people away and more towards truly understanding your audience, identifying great ideas, developing processes for gathering information, and building relationships with product teams.
Customer support might still be your primary goal, but insights implemented is a great secondary goal.