Yorick main here, here is the thing about his current pushing on towers
There have been two huge changes that help him recently between Demolish and Last Rites dealing the bonus damage on turrets
The damage on Last Rites is 30 / 55 / 80 / 105 / 130 (+ 40% AD) and using my build as an example (
)(36 from overloaded cdr) I end up with 377 ad and a base AD of 221 so IBG is dealing an extra 221 and a rank 5 Last Rites does about 820 damage before Armor mitigation
Meanwhile there is Demolish, which after 4 seconds makes your next auto deal 30% of your health to a structure which for this build is 1113 (plus some bonusdue to overgrowth, I used custom tool and didn't farm)
I tested 3 different hits on a tower, one with Demolish, one with Last Rites, and one with both combined just for the hell of it
So the Demolish hit did 1027, the last rites hit did 563, and the combined hit did a whopping 1300!
And these average towers are at 3800 health, so even if I had no cooldown on both Last Rites AND Demolish it would take 3 hits. And a majority of his damage on Last Rites comes from the fact he stacks it with a Sheen item and Sterraks to get a bonus chunk (Sterraks gives 71 bonus base as Yorick at level 18) so all in all you do get some respectable damage from Last Rites in the long run, but Demolish does way more in burst if you can't take the tower then.
And then secondly, don't let Yorick free push, that's why Last Rites does a ton, he gets these towers without being pressured