Swingers are the deal for Tree work. They have everything you are looking for.
But a skid steer is a widely versitale tool that mey help you make more money.
While tracked machines float, you cant turn without "shearing" off the grass in most cases. And they are expensive to take care of. One track is around $2500 to replace if you tear it or wear it out. A company I used to work for spent $9000 to rebuild the tracks after around 500 hours and that was just rollers and such, not tracks. Cat track machines are not all terrain. Many brands aren't. All it takes is some rocks to tear them up.
With your budget I would find a Bobcat A300. Tires are easy to replace. It turns. It picks up a great deal of weight. And has plenty of power for any attatchment.
I have a system I use to preserve grass.
1. Try not to go over the same area twice
2. make long sweeping turns
3. If I do have a tight yard to work in, I select one spot and do all of my turning there. Then only one area is damaged. It can usually be stiff raked.
4. Make the fewest trips possible
5. Full throttle is not necesary, 3/4 is plenty, The machine operates smoother,saves fuel, does less damage
6. If possible, drive straight in and back straight out
No matter what your buy though you have to be careful on grass.
Sure I have repeated some things said but I didnt read through everything