Good information in replies to your original post.
TRACTORS are designed primarily to PULL. You can PUSH with a tractor but the process is only semi-satisfactory.
SKID STEERS (wheeled and tracked) are designed primarily to PUSH. PULLING with a skid steer is really not practical.
Ground compaction is a function of tire contact area and equipment weight. Most compact tractors are sold with Industrial Tires these days, which are wider and less ribbed than traditional ag tires. I drive my Kubota B3300SU, 33-hp / 1,800 pounds, with Industrial Tires, across my St. Augustine lawn in Florida regularly without ill effect.
Turf Tires have tread, no ribs, and are wider than Industrial Tires. (Available much wider.) Turf Tires, especially extra-wide Turf Tires, would make the tractor feel less tippy. However your ability to pull is reduced with Turf Tires, you need
ribs on the tire rubber for
traction on dirt. And you would need wider implements which normally sized as wide or wider than outside width of rear tires.
FEL ("bucket" ) something I wanted immediately. The B3300SU is a Kubota Tractor/Loader package; I use the bucket all the time; usually for lifting/transporting logs and loading dirt into a trailer. That said, it is difficult to PUSH dirt smoothly with a bucket. I own a rake and a rotary disc harrow which I use to level and with these tools I can leave a smooth finish, whether is is woodland trail maintenance or preparing a yard for grass. I used all three of these tractor tools today.
There are a multitude of bucket accessories for the work you envision. I have a Bucket Solution's Bucket Spade and a set of aluminum Debris Forks from Payne's Forks. Both are very satisfactory on my B3300SU; however I am I would not recommend them for a small/lighter Tractor-Loader.
Before you have your tractor long you are going to want to buy one or more implements, most of which are carried in the three-point hitch. (3-Pt.) Whenever I need to pick up something sufficiently heavy with the FEL to possibly make the rear wheels light I attach an implement to the 3-Pt. as ballast. For me that heavy load typically is a tree trunk chained to the bucket and the 3-Pt. rear ballast is a Wallenstein BX42 PTO-powered
chipper, which weighs 425 pounds. Personally, I am not a fan of ballasted tires, front, rear or combination. Once the tires are ballasted you are always carrying the weight/compacting soil. With an implement you carry ballast only when required.