Hi Nate,
Without out a picture of your antenna I'm hesitant to guess what you have. If as you say it was installed in the 70s when KSL's primary signal was on low-VHF channel 5, you probably have an VHF only or all band antenna. This would mean it's wide and long.
Today you have no need for any VHF support. The closest you come is K08JR, which while information is sketchy on that signal. It does report being tied to KUTV CBS from Salt Lake. K08JR is rendered moot though by the fact K26GH-D rebroadcasts KUTV as well with a stronger signal at your location. Thus, according to your plot, if I were installing I would go with a UHF only design.
Question: When adjusting your existing antenna were you pointing AT Salt Lake (magnetic 245) or magnetic 330? Weak signals tend to vanish during the day. This is due to the atmosphere often being more tranquil at night meaning less resistance for weak signals.
If you are pointing at the Salt Lake Cluster, I would abandon that and focus on magnetic heading 330 translator cluster.
Magnetic 330 offers you via digital translator Salt Lake ABC, CBS, CW, PBS and NBC. FOX appears available via analog K40FY.
RE: Tree swaying. Sure, trees do sway. It may be a minor issue. ADTech may have some thoughts about this. I've never read anyone having issues with an a tree mount losing signal due to swaying.
Another option is a free standing tower. How tall are the trees around your cabin?
RE: replacement antenna. There are two ways one could go with this. If I had a tree mount (or tower) and was above the trees with an unobstructed view, I might install a Antennas Direct 91XG orientated at magnetic 335. The 91XG's precise beam and gain would get me the most signal into my coax. Plus, if you were mounted on a tree, a preamp would be a good idea as well to offset the likely longer coax run signal losses. Another fine option would be an Antennas Direct DB8e if you had to mount lower and had only partial pathway blockage. The DB8e's wide surface gives it the ability to "follow" signals that may get reflected due to blockage.
I run a 91XG orientated to magnetic 140 for a CBS translator 55 miles away, 7.9 db signal strength that broadcasts with 400 watts.
91XG:
Cheers.