First and foremost - to definitely answer your question about hunting cattails:
per SD law:
41-23-15. Walking, wading, standing, or operating motor vehicle on bed of nonmeandered lake,
or trapping or hunting on frozen surface without permission prohibited.
Second, one needs to understand that SD has three types of publicly accessible water.
1 is "navigable waters." South Dakota has three federally defined navigable waterways - The Missouri River, the James River, and the lowest 6 miles of the Big Sioux River. Federal navigable waters have public riparian rights including recreation i.e. usage of land below the ordinary high water mark to access/portage, etc. South Dakota also has a number of "navigable" streams and rivers as defined by state law. No lakes or sloughs to my knowledge are listed at the state level as navigable. State navigable streams and rivers follow similar laws as federal when it comes to public riparian rights including usage of land below the ordinary high water mark or above the low water mark.
2 is Meandered Water - generally this is lakes that were surveyed for statehood that were over 40 acres in size at the time of survey. There are around 300 meandered lakes in SD. Generally speaking, the public/state holds ownership of the lake bed below the waterline. Land owner riparian rights go up and down with the water line to the high water mark. Above the high water mark, technically the land owner retains riparian rights even when that land is under water, but the state does not enforce tresspass in those situations.
3 is Non-meandered water - all other water bodies fall under this premise. The water is still part of the public trust, but the land below the water is held by land owners. Could be an individual, could be state or federal (GPA, WPA, etc.)
Being as no lakes or sloughs in SD are defined as navigable per state or federal statute, there is no provision for public use of private land above the waterline regardless of the elevation of the water in respect to the high water mark. Private land is private land in SD. On meandered water bodies the public has a right to use the lake bed so long as that land is submerged. I.E. I cannot walk the beach for a mile to get to a better fishing spot if that land is privately owned, but I could wade in the water, for a mile parallel to that beach and be 100% legal.
On non-meandered waters, the public has no right to recreate on the lake bed. The public has the privilege to recreate in or on the water i.e. boating, ice fishing, waterfowl hunting from a boat, etc. Anchors, lures, etc are considered incidental contact and allowed. If your feet touch the land under the water and that land is private land, you are tresspassing. Under current law, land owners with water over their private land can legally close that water to recreation.
Some SD water history - when the glacial lakes/prairie pothole region started to fill up from wet cycles in the 80's and 90's, many small sloughs exploded in size, eventually flooding hundreds of thousands of acres throughout SD. I'm sure many of you have seen scenes caused by this flooding first hand, via hunting or fishing videos, pictures, etc. of flooded farmsteads, people ice fishing next to flooded farm equipment, and so on. Many of these bodies of water flooded roads or state/federal property, thus allowing a means of public access to the water bodies for fishermen. The state GFP started stocking these water bodies with game fish; the lakes were super fertile, fish growth rates were phenomenal, & a fishing tourism industry exploded. Word would get out of a hot bite and there'd be 50 trucks lined up to back boats in off a road grade or parked along the shoulder and every field approach for 1/2 mile. Many locals got tired of the traffic, litter, noise, etc. Around that time some families with flooded private land sued the state GFP stating the public should not have access to this type of water. They won and the bodies of water their land was beneath was placed under an injuction - the GFP nor the public could facilitate recreation or recreate on those water bodies. Fast forward through the years and many more land owners were fed up with littering, noise, blockage of roads, tresspass, etc. and additional court cases were heard. In 2018 the state supreme court ruled that the public does not have the right to recreate above flooded private land and the GFP closed boat ramps on 18 non meandered water bodies. A special legislative session was called, and a compromise bill was passed that gets us to where we are today - land owners have the legal means to close water to recreation (fishing) over their flooded private land.
Long story short- you may only hunt frozen surface water cattails over meandered water below the ohwm, navigable water below the ohwm, or over non meandered water with permission or above state/federal land.
Here's a few links I've bookmarked from years past when this was an ongoing political issue.
https://sdlegislature.gov/Statutes/41-23
https://gfp.sd.gov/nonmeandered-waters/
https://www.sdpb.org/politics/2018-01-18/non-meandered-waters-resurface-in-south-dakota-capitol
https://congressionalsportsmen.org/...ssion-passes-nonmeandered-waters-access-bill/