I am so sorry about your rabbit, and no you are definitely not a wimp, you are just a very caring person, who loves their animals.
I have had animals all my life, and they mean the world to me. I always dread it when they get ill, or old and have to be taken to the vet for the final time. I do still get upset, and probably the worst to deal with is my horses. At the beginning of the year I lost my oldest horse who I had on the farm with me for 32 years. Having her shot when she got down was hard.
But I know that my animals have had a long and happy life, and I always think that this is the last thing we can do for them, hard as it is for us, its the kindest thing for them.