The other two answers address the "rolling" scenario quite well, but I'd like to point out one glaring error you made in your reasoning: the Earth isn't rotating along the Y-axis. It's rotating on an angle.
Now, of course, you know that. But the important point is that this causes the seasons. If Earth was rotating exactly around the Y-axis (with respect to the orbital plane), there would be no difference in climate due to orbital position.
As you let the rotation axis fall away from the Y-axis, you get more and more seasonal weather, up to the point where the rotation is aligned with the orbital plane (it doesn't matter in which direction - that's just an offset to when each season starts, but the effect is the same).
For additional points, changing the rotation rate can give you more to work with. If a day takes half a year, it's going to have a huge effect on the seasonal variations - it might be interesting to explore a scenario where the rotation is a bit more tilted and slowed down. Venusian weather and climate is pretty interesting, in part due to its slow rotation and massive atmosphere; while the planet's day is longer than its year, the clouds make the trip around in measly four Earth days.