Actually no, it’s getting a bit into physics and seems super counter intuitive, but pushing to the side actually generates a lot more forward velocity than pushing straight back regardless of what method you compare it against. The reasoning is kind of what you touched on, because the force along the y axis (the outward part) is forced into forward velocity due to the sliding skate only being capable of forward or backward motion. It kind of operates like a gear box, it requires more force but you get a lot more velocity out of it compared to the same amount of force applied straight backward. It’s why you’ll see players strides widen as they get up to top speed, the same way you’d start a car in 1st gear then shift up at higher speeds.
If you’re a nerd like me and this at all intrigues you, this topic is a lot more covered around sailing. It’s the same concept that let’s sail boats travel faster than the wind speed powering them. A weird way to think of it in a simpler way is imagine trying to grab something slippery, and how it shoots out of your hand. Now what moved faster, your fingers while they were squeezing it or the object that shot out, despite the direction of the force almost being perpendicular?