It's rare to see that much consensus here which tells you something. Change as little as possible in terms of the parts, overhaul the bike and replace the consumables is the best course of action. The 14-28 shimano freewheel you listed is a good one if you decide to change out the freewheel. You might want to try dripping some oil down into the freewheel guts (spin it and you'll see the areas where oil can get in front and back) to see if that cleans it up. Everything likely is in fine shape and rebuildable.

That is a worthy project and a great color.