Logged areas often have issues with erosion or at least loss of nutrients in the soil as sun/rain is now reaching the soil directly rather than being filtered through tree foliage. Cut areas are also excellent habitat for poison ivy, asian bittersweet vine, honeysuckle and other sun & heat loving plants. Planting into the landscape with fast growing seedlings (birch for example) and cover crop-type seed mixes would help hold nutrients in the soil and begin to build shade to keep back the unwanted vines and such.
You do want to clear the streams I think. Though natural tree falls are part of stream ecology, you don't want the stream choked off by stuff left by the loggers. You can rent a wood chipper and build up some piles that can be used later for trail building and spread around the area to hold soil and nutrients. When we bought our property, the two house sites and the drive were covered in the pine wood chips from clearing the house sites. I thought that stuff would never decay. But after the guys scraped it all back and piled it up on one of the house sites, it has decayed nicely into some really good looking soil.
Our neighbors land got logged recently, and they took out all the old white and red oak. Granted some of the limbs get relatively thin, but these were big old trees with large crowns so there is burnable stuff (at least to my eye) almost all the way up, especially in our Morso's rather small fire box. However, it is on our neighbor's property. I'll have to clear our stone walls of some stuff that's fallen across the boundaries, but it seems like he's just going to let it lie there, unless he's just letting it age in place and will chop it up during the summer.
As far as brush, first you want to know what species are in there and whether there is anything you might like to keep. Our friends had a similar heap of vegetation that they were just going to have removed, but when they looked closely, they realized that while asian bittersweet vine was the dominant life form, there were a handful of stunted crabapple trees underneath. It was extra work to get rid of all the vines but now they have some crabapples trees that bloom pink in the spring.
If there's nothing workable in the pile, our landscape architect would recommend removal and replanting with things that would help you maintain the view by not growing tall and blocking everything. Create a bit of habitat with some shorter trees (witch hazel, musclewood) along the edges, no-cut grass or lawn substitutes (thyme), woody bushes (willow,) and so forth. The more native species you can use the lower the maintenance you'll have and the more predictable the plant behavior will be. Less overgrown areas may also help control tick population also.
Ernst Seeds is a really good resource. Our landscape architect uses them to create site-specific custom seed mixes, but they also make pre-mixes for soil recovery and protection in a lot of different situations:
Browse Seed Mixes | Ernst Conservation Seeds
Most states have some form of tree nursery program, and those usually sell seedlings to the public sometime during the year to encourage reforestation.
This is the one for NYS.
Vermont has one of the best resources for willows on the planet. Really good for erosion control, soft fencing and habitat creation.
Vermont Willow Nursery
Edit: Forgot this - NYS has a "Private Forest Management" program through the Department of Environmental Conservation. I'll bet that Vermont has a similar program:
Private Forest Management - NYS Dept. of Environmental Conservation
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