Weepy hagiographies are not my style. Besides, living on a farm for a few summers taught me not to elevate animals, even pets, to the status of humanity. I believe that we can celebrate the joys of having pets, and also mourn the loss of them, without straying into anthropomorphism. Family, friends, and strangers are more important than pets. That said, I've had a pet — or at least cohabited with someone else's pet — for three-quarters of my life. I do miss them. I'm grateful for what they brought to our lives, and I'm satisfied that we were good masters for them.
The joys of having a pet may resume someday, but that's a matter to be explored thoroughly beforehand. Within the expected lifespan of a puppy or kitten there is certain to be some kind of discontinuity in my life. For example, I doubt that ten years hence we will be living in this house with its ample, fenced back yard. We do like to travel, and in arranging travel I think we would enjoy a degree of spontaneity that pets have often precluded. In time we will sort this out.
For the record: although definitely a dog person, I had a great relationship with the cat, Phantom, whom we buried at sunrise. I worked to forge that relationship, and I think he did too. I don't expect such a relationship to recur, but I never expected it to happen once. Life has its pleasant and rewarding surprises.