After looking at a photo of your model, there are three things that are obvious. All are design issues and have nothing to do with how it is operated.
Number one design issue is that there is not enough weight over the drive tires. The correct design should have the engine somewhat behind the rear axle, while driver and mowing deck is ahead. This is why when you put your seat back, things improve - you are shifting weight rearwards. I don't know the correct ratio, but if 75-80% of the entire operating weight (with driver) were over the drive tires then the drive tires should be able to move and steer the unit properly.
Number two design issue is the distance between the drive tires. If you consider the 4 tire arrangement on the ground, it looks to me like it is shaped like a rectangle - the longer sides are front-to-back dimension. It should be the other way around, or at least square. The rear tires do not have sufficient leverage to turn, even go straight in conditions such as a slope.
Number three issue is the tires. They are too smooth for the work they are expected to take on.
The big question is what to do. If you cannot return the unit and get a better design, then I see a couple of options. First is to consider more aggressive tires, just as you pointed out. If there is a concern about tearing up grass by circling a tree, then don't circle it. Slice/mow the circle 25% at a time, and don't pivot on one tire anywhere in the yard. Another option is to add spacers for the rear tires to put more distance between the tires, thus creating more leverage. This would only work if you can find a way to do this and the width is not greater than the mower deck. The final option I can think of is to make a bracket on the rear of the frame to add weight. Sure, this is some customization where someone can be creative with how it is achieved. The intention is to shift weight rearwards so a greater percentage is over the rear axle. By adding behind the axle, weight comes off of the front tires.
