I own the same book and was similarly surprised when I read that instruction, but in the section on ingredients, the author does mention a particular variety of onions called pink onions. The mention is on pg. 32, and there is a picture of a pink onion on the upper left of pg. 34.
Here's an excerpt from pg. 32:
"The longer the onions are fried, the browner they will get and the deeper the color of the curry will be... When the onions are fried until only light pink in color, they will impart a sweetish taste to the curry. Certain varieties of onion, like Spanish onions, are too sweet to be appropriate for curry-making. The most appropriate from the taste point of view are the French and the small pink English."
From 50 great curries of india
I've never seen pink onions in any of the local markets (in northwestern US), I used white onions and sauteed them just to the point of browning. The recipe turned out fine, but it's a lot of onion. (The recipe calls for 2 large onions, finely chopped. It's a recipe that produces 2 servings.) Personally, I'd go a little lighter on the amount of onion or use yellow onions (despite the author's recommendation against Spanish onions), but that is just personal preference from a Western palate.
Note: it looks like the current printing of the book is different from the one I own, so the page numbers I cite may be off, but if you search inside the book (on Amazon) for "pink onion" you can see the page I've cited and the picture of what the author calls a pink onion.
Of the recipes you link to only one contains paprika, another one lists "Chinese chili paste, chili oil, hot sauce or dried red chili flakes to taste". The one that specifies paprika doesn't mention a specific type of paprika. Without further distinction, that generally means the sweet, non-smoked variety. Sweet paprika is just dried, powdered pimento, the type of pepper seen most often (here in the US anyway) stuffing cocktail olives. It's not hot at all. Hot paprika usually gets its heat by mixing in another, hotter, pepper. Even those paprikas are not generally overwhelmingly hot.
Paprika of any type (hot, sweet or smoked) from any country (Hungary is known for it, as is Spain) is going to bring you closer to a flavor profile more reminiscent of borscht's Eastern European roots. You may want that, but I wouldn't count on paprika to to provide the kind of heat you're talking about.
For Chinese hot and sour soup the best sources of heat are white pepper and chili oil. Nothing else tastes "right" in my opinion. With that in mind I lean toward those ingredients for your borscht as well. You can buy chili oil in any big grocery store, but Asian markets will have a greater variety at probably a much lower price. You can also make your own.
So I'd recommend starting there. White pepper, chili oil and paprika. Make a batch of soup with none of the above and add each spicy ingredient to samples of soup until you hit the combo you like. For best results, once you got the ratio of spicy ingredients figured out, scoop out a bit of your broth and simmer the spices for a good 30 minutes or so (the pepper and paprika especially, it doesn't matter so much with the oil), and add the simmered spices into your cooked soup. They're just better cooked a while. Next time you make the soup you can add the spices in the beginning, since at that point you'll know what you like.
Best Answer
Sauteed onions can provide both caramel flavors (from the sugars in the onions) and Maillard reaction compounds, depending on how they are sauteed. Thus onions can supply a range of "umami" flavors for soup which otherwise you need to get through roasting animal bones and other tissue (e.g. brown veal stock). Of course, even beef stocks often add onion as well for extra flavor.
As an extreme example of this, I often prepare a vegetarian French Onion soup using a meat-free broth made entirely from onions, onion skins, and cheese rind. Blind tasters often fail to distinguish it from a store-bought beef stock.
You can get a lot of the same flavors from combinations of other browned vegetables, but onions neatly supply a perfect package of flavor compounds in one inexpensive, long-keeping root vegetable. Why use anything else?