Two things control the "sharpness" of onions: variety and age. While certain varieties of onions are sharper than others (i.e. Reds, walla-walls and vidalias are sweeter), any onion which has been in storage too long is going to be sulphurous and sharp-tasting. Since it's January now, that's going to be pretty much all onions.
Since onion sharpness comes from sulphur compounds (as I understand it), the best way to sweeten onions is to release some of them ahead of using the onion. The best way to do that is:
Slice the onion thinly, crossways (parallel to its equator rather than pole-to-pole).
Place the sliced onion in a wide bowl, uncovered or very loosely covered, in the fridge or other cold place for at least an hour and up to a day.
Additionally, either vinegar or salt -- or both -- will help accelerate removing sharpness from onion. For example, for a New Year's Eve appetizer this year I sliced a red onion in to rings, tossed it with 1/2 tsp salt and 2 tsp red wine vinegar, and left it in an open bowl on the 45F porch for 5 hours. The result was intensely sweet red onion which could then be used as a canape topping.
If you're in a hurry, though, the only way to make the onion less sharp is to cook it.
Paneer butter masala, malai kofta gravy and matar paneer are part of the North Indian cuisine where the base is mostly the same with some minor variations (like excluding cumin seeds).
So essentially, you have a common base gravy for sweeter dishes like these and then you have a spicier gravy for other dishes.
Many Indian cuisines have this pattern. For example, Gujarati cuisine will have red chili powder, turmeric, garlic-ginger, mustard seeds, cumin powder, fenugreek, cilantro and jaggery, etc. as a general base with some additions and subtractions of the vegetable ingredients.
Similarly, you'll be able to find common base gravies across Indian recipes and cuisines.
But, when you say:
"In addition - It seems to me that the same thing could be said about aloo gobi, chana masala, baingan bharta, and many other dishes - Basically the same frying of onions, ginger garlic paste and tomatoes, but with a different ingredient added at the end (for example eggplant for baingan bharta)."
You are making a generalization which is not necessarily true. These recipes can vary quite a bit based on regional tastes, specially baingan ka bharta. This dish is a signature dish and needs a seasoned hand to deliver it right.
So, you are right when you say there is a common pattern across recipes.
But certain dishes like aloo gobi will exist in different forms across the country. But rest assured malai kofta, paneer butter masala and matar paneer are only prepared in the North Indian style.
Best Answer
Adding fried shallots or caramelized onion may help.