I didn't say that it's not hurtful, but you did say that animals die quickly. I was just pointing out that not every animal dies that quickly.
Since you did ask...
Refusing your future kids the opportunity to have pets just to protect them from loss will likely do them a disservice. Kids need to learn how to cope with death. Are you also planning to refuse your kids to know their older relatives because the death of a grandparent would be sad? What sort of coping mechanism do you think they would have when they are adults if they were protected to that extent?
I'm fairly certain that if you asked your girlfriend, she'd much rather have the happy memories with her dog than to have none. Or maybe she needs to be nudged in that direction.
If your actual reason for not wanting pets is not wanting the responsibility (or thinking that the kids are too irresponsible) then that makes sense. The shelters are full of pets that wound up with irresponsible loser owners who didn't think things through.
Why do I always do this to myself???