IMO you need to choose a replacement ground cover. Bare ground that receives rain will eventually grow something.
On my property it is the little tiny seeds from invasive Siberian Elms. I hate those trees!

However, the riparian areas that I have reclaimed and established tallgrass prairie work very well to suppress that tree growth. (Obviously, tallgrass prairie will not work for your application.)
You might ask your county extension agent for something that would help achieve your requirements.
As regards herbicides, glyphosate (old RoundUp) is a very effective general killer. Once it hits the ground, it will electrostatically bind with any clay particles and become almost completely inactivated. It will stay bound even if those clay particles subsequently wash into the pond.
Just make sure you don't spray directly into the pond. I would stand at the pond's edge and spray strips of vegetation when a light breeze is blowing away from the pond on your spraying direction.
I am certainly NOT genius enough to come up with perfect solutions to complex problems at my farm. I have all kinds of areas marked with "test projects". Unless you need everything killed yesterday, then I would try some test strips where I treated and then planted a replacement ground cover. You can experiment with different ground covers and see which works best for your soil and climatic conditions. (Unless you MUST have a uniform look around the pond. I sometimes do get into short-term trouble with my wife!)
You could even just create some walking pathways using glyphosate. I do this on some of the wetter areas of my farm where I usually can't mow. I make walking paths early in the year after most of the earliest emerging plants are grown enough to be nuked with the herbicide. Just people walking down the path keeps it "semi-cleared". I then have to spray again later in the year when the warm-season grasses and summer weeds have emerged. You might be able to do this around your pond with a 2-gallon hand sprayer. You can even spray right up to the water's edge in the best fishing spots. (A tiny bit of glyphosate in the pond is not going to hurt anything.) Do NOT make a path straight down the slope to the pond. If that path becomes bare dirt, the rains will erode it into the pond. A few meanders across the slope will prevent that problem.
Other options would be to re-grade your pond edge slopes with a dozer or even a road grader. That sounds like a lot of work, but a good dirt mover could probably knock it out in a single day with a grader. Do the farms in your area have terraces to reduce erosion? I think an operator that does terraces in the area could give you a reasonable bid for your project.
Another option is to purchase a side-boom mower for your tractor. (They may not make one for a really small Kubota?) With a side-boom mower the tractor can stay on the flatter ground and then reach out to mow the steeper slope with the boom extended. Our county guys use them to mow all of the steep-sided bar ditches adjacent to the roadways.
If you can't afford that option personally, there may be some "rural" mowing services with a side boom where you could hire those guys to buzz your pond a few times a year.
Good luck on your shoreline improvements. A pond that can't be reached by your family is not much fun at all.