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He has never bitten me or anyone before in his life, except for play bites of course, but the image is stuck in my head, since I know large animals can be unpredictable.
That's not exactly true.
Firstly, most large dog breeds have extremely gentle temperaments (they've been bred this way). They actually recommend getting a large breed if you have young children, because they are very tolerant and kind and are
extremely reluctant to bite people. I'd be more afraid of being bitten by a Chihuahua than a Great Dane, personally.
Secondly, dogs give clear signals before they're going to bite, and if you know dog body language, you can see when a dog is likely to bite.
This link describes what body language dogs give when they're feeling certain moods. They don't bite randomly. If your dog is giving off a mix of friendly and submissive body language, he is
not going to bite.
Now, many people don't know how to read dog body language. But if you, as the owner, can read your own dog, you can tell when it's safe for others to pet your dog and when it isn't. When someone asks if they can pet your dog, look over at your dog's body language to see if they seem to be in the mood for petting. Excited and/or friendly behavior, especially when mixed with submissive signals, means it's safe for others to pet your dog. (I still remember this one dog who, when I asked his owner if I could pet him, he immediately came running towards me, showing very friendly/excited body language. I knew immediately that the
dog at least consented to being petted!)
I have a question for you. Do you think maybe, once your current pets have lived out their natural lives, it might be good for you to not get any more pets? If you don't have pets, you can't feel anxious about their welfare.
Personally, I love having pets, and while I'm pretty sad if anything happens to them, I don't tend to worry about them much. My cat goes outside regularly, and while I know it's possible she could get hit by a car, injured by another animal, etc, I trust in her judgment to keep herself safe. She's a pretty smart cat, she knows about avoiding cars and she's good at figuring out who she could or couldn't take in a fight. (In fact, if all the humans of our family died, as long as she managed to get outside, she'd survive just fine. She'd be sad, but she could easily take care of herself. She's even a good mouser.)
I did get anxious about my rats because a few times I came to their cage to find one of them had suddenly died. But never to the extreme you're talking about, it's just that if I went more than half a day between checking on them I'd get anxious, and when I was heading to the cage I'd always get this little nervous thought that one of them might have died.