You are correct that you absolutely need some plants surrounding the pond.

They stop erosion on the slopes to the pond and tend to catch any silt that is suspended in the surface run-off water after heavy rains.

They also slow the incoming surface water, which greatly increases the infiltration rate of that water into your soil. That is the process for keeping some of the dissolved nutrients out of the pond.

(Judging by the language in your post, I am pretty sure you already knew all of that.)

Addressing what happens to the nutrients when your plants die back in winter, is highly dependent on what type of plants you have in your buffer zone.

If you have shrubs or trees that shed a lot of leaves or needles that can easily be blown into the pond, that is a net negative.

If you have grasses, they will largely decompose in place. My native tall grasses form a natural thatch over the years. We try to burn every 4-5 years to combat the lost productivity due to the build up of thatch. (We don't burn any of the grassed waterways leading directly to my ponds.)

Likewise, you can probably observe the build up of thatch in the parts of your lawn that you mow. However, a good mulching blade will help get most of that into small enough bits that it will decompose in place, rather than forming a thick thatch. (I don't know if you mow around your pond.)

I believe that you have surmised that a heavy rain in the winter will actually float some thatch (and nutrients) into your pond. However, I think that is usually a minimal transport of nutrients.

IMO, if you want to go for maximize nutrient exclusion, I would mow and bale (tall grass) or mow and bag (short grass). If you are not selling the bales though, that is a lot of work for only a little benefit.

If you want a partial solution, then you could mow some strips perpendicular to the slope and leave unmowed grass strips in between to catch any organic debris before it enters the pond. This is similar to the "silt fencing" you see around construction projects where the work created bare ground.

This method does look goofy, especially if your wife likes looking at a well-manicured pond setting from your deck. However, all of the wildlife on my farm loves having mowed areas in between tall grass areas. The deer, turkey, quail, and even the coyotes love traversing the property in the short stuff, and then as soon as they sense any danger they fade into the adjacent heavy cover.

I don't know how much of that long reply actually applies to your situation? Hopefully, you can find some little bit that actually is helpful to your pond management situation on your property.

Have fun keeping your pond in good shape!