What are your thoughts on a tractor versus skid steer?
Both are good of course if you have unlimited money.... There are more low cost implements for a tractor and it's a lot easier to find them used, and tractors are generally cheaper to run and fix as well.
Skid steers have more hydraulic flow and pushing/lifting power so they can run some implements a tractor can't easily (I'm generalizing a bit here..) but things like a heavy forestry shredder on the front end of a skid steer is just going to work better/faster than a comparable tractor implement. And yes your dad is right you can push more dirt and lift more weight with a skid steer.. but again .. does that actually matter for your use case. Skid steers are also more agile, which can be pretty useful for things like cleaning stalls (until someone gets the thing somehow stuck in a stall so bad you have to take apart half the barn to get it out... but i digress..).
If you needed to push in small roads, or move a lot of material.. then.. a skid steer might make sense. But from what usage you've described the cost/benefit tradeoff is more towards a tractor.
brush hog versus flail mower
Again it's all trade offs.. The flail will leave a nicer cut path in grass areas, doesn't take as much room/stick out as far behind the tractor, and technically you can usually get a touch wider cut with the same HP, you can also get one that will offset a bit to the side of the tractor which is nice for fence lines. OTOH brush hogs are tough as hell (at least the good ones), pretty simple and generally fairly cheap to maintain (new blades are cheaper anyway... and you can sharpen them a few times as well..) and you could get a good one used on marketplace or craigslist for under a grand in the size you'd want. I rarely see used flail mowers, basically never small ones especially (there are some ag sized ones I've seen which whooo $$$$$), and a new one is $3k+. I think brush hogs are a little more forgiving in rough terrain as well. So.. yeah.. "it depends on budget and use case".... For you... probably a 48" brush hog and you're good to go.. or maybe a small flail mower . Look for one with lots of cross bracing, thick metal, and a decent looking gearbox.
question about brambles
Once you have a tractor you can start wrapping a chain around them and ripping them out. Then once the young delicious shoots come up in the spring put some goats on them.... then you'll spend all your time chasing after and yelling at the goats and will forget about the brambles entirely...