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Thanks for the ring-barking idea - that sounds promising. (I know of that as "girdling", but hadn't thoguht of it.)
As for just putting a metal collar on the tree to keep the squirrel sform climbing it, I do hate to cut it down, but a collar wouldn't solve the problem.
For one thing, its branches are very close to the power lines, which the squirrels use as a highway system, and pruning wouldn't fix that.
For another, the squirrels have parties in the tree - five or six of them will hang upside down eating the samaras (helicopter-shaped seed pods), and then they leisurely scamper all over the roof , exploring any holes that already exist and making new ones simply because they're there and they feel the need to gnaw (squirrels need to wear down their teeth on something). They sit on the peak, hang their heads over the edge, and make a hole in the fascia board. Or they start digging in the valleys of the roof and get in there.
Last winter I trapped and killed four squirrels that were living in the eaves. (It's illegal here for homeowners to transport squirrels off their property, in part because that can spread diseases; they come back "home" anyway if transported up to 20 miles or so away; and if you take them farther away they tend to invade somebody else's house because they are "rogue squirrels" that have learned to prefer houses to trees in the winter.) I don't want to go through that again - I'd rather kill the tree than more squirrels.
But I might put a metal collar on the stump (once it has a platform on it), in case they still like it and use it to get up on the roof a lot. The point is that they can get onto the roof with or without the tree - they can use the power lines or downspouts, and I'm pretty sure that I've even seen them go right up the brick wall of the house in the past - but the tree's samaras make it the favorite neighborhood hangout joint.