That is such a wonderful way of easing the transition for you, and, as you say, extending the life of the older one.
I imagine that would work with dogs. With cats, the “break-in” period of introducing a new cat can stress out the older one, and actually make it worse.
For my previous two (before the two I have now), the first one died just two weeks before the other was diagnosed with what I was told was an aggressive cancer. I was told it was a matter of months, so instead of spending her final months trying to adjust to a new kitty, I decided to let her live out her remaining days with just me, free of stress, and then I would get two new ones, together.
It turned out that she lived four good years after that, and she wasn’t uncomfortable until the final week (which I helped end). The vet told me that the number of kitties with her stage of cancer, at diagnosis, who live four more years is less than 1 in 1000.
I like to think that she loved me, and the home I gave her, so much that she wanted to stick around.