Actually, I have thought it out. I am an experienced horse-packer and know goats (and horses) VERY well! I am well aware of their health, temperment, and training issues. I've trimmed many a goat foot. I wouldn't even begin to do this without a well-trained goat or two -- whether I trained them myself (by being there long enough to do that) or by buying (or hiring) trained pack goats. As for strength -- I agree that strength has nothing to do with control of a horse. The exception is in a panic situation (a bear or cougar) in which even a well-trained pack horse may try to bolt. A goat, even if they do panic, can be held by strength (using a belay method). Traffic isn't any more of an issue with a goat than a horse -- if they are trained and accustomed to it, they are fine.
Goat packs (in many sizes) and all the equipment are readily available in the US (really!) as are resources (books, listserves, etc). So it's not as crazy as it may sound. I have trained horses, dogs and, yes, goats (although not for packing yet). In fact, I have a PhD in animal behavior, so I'm pretty savvy about the pros and cons.
A trained goat happily fits in the back of a car (yes, I've done this!) although they aren't potty-trained
As for the dog issue: I would carry a battery-charged electric fence and stake the goat(s) in it overnight. Pepper spray and horned goats are the best protection on the road.
Thank you for your concern! But I believe I have enough experience to pull this off. First step: this summer I'll be doing about 10 days on the Camino with my daughter. That will give me a better feel for goat-constraints. And I'm checking out goat packers in Europe (there are some, it seems).
My biggest concern will be with finding a good home for the goats while I'm not in Europe. I certainly won't acquire any until I figure that out. (Too many goat meat lovers out there....and when I first had goats I lost one to a dog attack in the yard).