Ecclesiastes 3:3. (there is) a time to kill and a time to heal, a time to tear down and a time to build.
A few years ago, our 16 year-old cat, Charles, developed kidney problems and his body could no longer process the protein from food. He grew thin and was in pain. I took him to the vet and held him as they injected him with a lethal dose of aesthetic. I shed a bucket load of tears. He’d been part of our life since he was 5 weeks old. I vowed that I couldn’t go through that again. No more pets. I thought I was going to miss him forever. 
Over the coming year my grief did heal, and about 18 months later I decided that I wanted another cat. So off to the Animal shelter I went, and come home that day with Revv. He’s been an absolute delight and I have no regrets about bringing another fury friend into our lives.
Time really does heal if you let it. Healing can also be a matter of opportunity. If that opportunity presents itself, grasp it with both hands (or head and heart) and move on.
It’s rough when they die. We just lost one after 14 years. Left an empty space inside me. I still feel pain, after a month, keep expediting him to appear, in all those favorite places. I’ve had relatives die that stressed me less. They didn’t live with me every day and night for 14 years. Damn I miss that cat!
I know what you mean about your family comment. Been there too… funny, but not funny! After 6 months I still look for my dog at the gate when I drive in. Glad I have the memories.