Peruna,
You can get 3 speed rear wheels, that is if you want gears.
Options
Sturmey Archer or Bendix vintage wheels on S2 rims or look for the a newer Nexus 3 internal 3 speed on a 2.125" wide rim. I'm going to just lace a correct CMC rim to a Monark/Whizzer HD hub with heavy spokes.
I think the 3 speed shifter was for the gadgets not the actual rear wheel, but if you want the gear it works out, thought do note the it is better to have a larger front sprocket than the schwinn clover for better gearing.
We actually have a DX cruiser setup with an s2/nexus and that from clover and the gearing is so-so, not like brand new cruisers.
Oh, and you can pretty much adapt that Shimano shifter to any rear.