Even if the steering wheel isn't trembling, the wheels could still need to be balanced. To be safe, I would start with a wheel balance and also inspect the suspension and tires.
When it cuts out like that, especially quickly, it's a terrible feeling. Well done for persevering and ultimately figuring out the cause. Hopefully, the new injectors will resolve the issue so you may resume enjoying the X5 stress-free.
For winter wheels, a square arrangement is quite acceptable and frequently superior for winter grip; you don't have to keep staggered. Depending on brake clearance, most people switch to 17" or stick with 18", but 17s with the right winter tires typically ride better and are less expensive. Just make sure that everything is cleared by the widths and offsets.