Why use onions in cooking?
Their primary purpose is flavor, although (as mentioned by @David Richerby) they can also be used for thickening in some cases. If you don't like them, but other members of your household do, then it becomes a cost-benefit analysis of every dish.

If your hatred of onions is visceral, yet other family members would just kind of rather have them than not, not including them is easy. It becomes trickier if everyone else loves them.

Consider prepping onions by quickly soaking the chopped onions in cold water or giving them a quick pickle in vinegar and salt. That may remove flavors that you find offensive while still keeping those qualities that other members of your household may like.

Garlic provides much of the same flavor profile as onions. Can you add a bit more garlic when you eliminate onions? 

You might find some types of onions less offensive than others. Consider supersweet onions like Vidalia, green onions (AKA scallions), and red onions that have been tamed a bit by soaking or pickling.

Consider asafoetida as a substitute, along with other substitutions found here: What can I substitute for onions?