Are you old enough to remember the bottled salad dressing that was called "French dressing" in America? I sure am. I suppose it's still available in supermarkets. It was big in the 1950s and '60s, when I was growing up. That was before I first came to France and realized that the real French salad dressing was home-made vinaigrette.
According to the Wikipedia article about French dressing, it was made with olive oil, vinegar, tomato paste, ketchup, brown sugar, paprika, and salt. Notice the ketchup and the brown sugar, both of which are not only sweet but have very strong flavors. I say: leave them out.
Recently, I've been making a dressing that I'd call vinaigrette à la tomate, or tomato vinaigrette. It turns out to be really good on salad greens. It looks like the old American "French dressing" but its taste is clean and fresh, not cloyingly sweet. The ingredients are tomato paste, Dijon mustard, white wine vinegar, salt, pepper, and salad oil (olive or other). Actually, for years I've been making tomato vinaigrette using finely chopped fresh tomato (in season), but not with tomato paste.
For the day's salad, spoon one generous tablespoon of tomato paste into a big salad bowl. Add a teaspoon of Dijon mustard, a teaspoon of white wine vinegar, salt and pepper to taste, and stir well. Once it's well mixed and the salt has dissolved, whisk in 2 or 3 tablespoons of oil. Optionally, add some onion, shallot, garlic or herbs like parsley, tarragon, or oregano. Toss salad greens in it. You'll like it. It's just a vinaigrette with tomato paste added to it.