I have a recipe for a Ukranian borscht.....very like what you say....but it begins days ahead by making a fermented kvass.
Other than that it's pretty well beef broth and beets.

linda 6, my Hungarian Grandmother and my mom made the best beet soup. I don't have a recipe as everything was done from taste. I made it quite a few times, but haven't done so in a few years, so I am going by memory here!

Cook 2-3 pork hocks in a soup pot filled with water and season with salt & pepper, bay leaves and some onion. Simmer slowly for about an hour or until pork hock meat is tender. Remove hocks and add sliced beets and potatoes, peeled and cut into chunks, season with salt & pepper and cook until vegies are tender. Make a batch of egg dumplings in another pot and add to the soup. Chop up the pork hock meat and add to the soup pot. Add a tablespoon or more of vinegar and sugar, temper about a cup of sour cream with the hot soup and add to the pot. If the soup is not thick enough for your taste, make a roux with flour and oil and add to the soup until a disired thickness.

Thank you all for your comments. I think one day I'm going to try this by taste. Hope it works.