Some linguistics/semantics:
In my language the word meaning vulnerability literally means woundability. Like, if you are vulnerable you can be wounded by someone.
So, it's just that. If you are vulnerable, other people can cause you pain. IME many of the most significant, important and even sacred things in life can only be experienced if one is willing to take the risk of getting hurt. Also, most if not all of the worst, most horrible and traumatic things in life include the risk, and also the fact, of getting hurt.
I personally try to deal with vulnerability (which is far from easy!!!) by thinking that I'm taking a calculated risk. I may end up getting hurt, but I think I'll survive it and I'm willing to pay that price. Let's continue the examples including animals. Okay, I want to have a dog. Chances are, I'm going to outlive any canine. A dog dying means pain. A lot of pain. Been there. So, by getting a dog I'm making myself vulnerable. I will get hurt. Is it worth it? Absolutely.
With people, though, it's more complicated. You can lose the person but also get hurt in many other ways. It's not merely about keeping/losing. I've found out that some really awesome, almost sacred things can happen between people when they agree to be vulnerable. Compassion and consolation are among them. Only when I open myself up in a very vulnerable way can someone truly comfort me. Carry my burdens for a while. It's not something I learned as a kid. In the world I grew up in things like that didn't exist. Everyone was on their own. It came as a total shock to me when I first experienced true consolation. Someone else took my pain and made it his pain for a while. It was - I can't describe it any other way - a sacred moment. And it definitely called for a great amount of vulnerability.
So yeah, daring to be vulnerable has definitely changed my life. And I've been hurt, many times. Still, I'd say, it's been worth it.