I prefer a tractor for you situation. My Massey Ferguson GC2400 mows very well with its 60" deck and covers ground at a pretty good pace. No it won't be as fast as a zero turn, but I prefer a one machine solution. IMO, unless you need a tractor that is too big/heavy to mow with, or you have a huge (think many acres) amount of lawn to be groomed, I don't think it's worth having two machines to store and maintain.
A smaller tractor will be better for mowing, a larger tractor will be better for ground engaging. Either should do most of your tasks well, with a couple of exceptions. First, moving 30 yards of material will take a while, the bigger the bucket the faster it will go, but the tractors in a reasonable size for mowing will still have a "small" bucket. I spread 30 yards of mulch this year with my little tractor, it was a LOT of buckets! Second, the trees. I assume you'd like to remove the stumps? You wont be able to do it easily. You won't even come close to being able to pull them out, and trying to dig them out with the bucket will create quite the mess. A backhoe is the best way to remove the stumps with a small tractor, but they are very expensive, and hard to justify unless you have many tasks for it. I would just rent a stump grinder.
The JD2520 you're looking at is a small CUT, the Kubota B series would be similar in size/capability.