Hmmmm. I would not do it, but I hold the unpopular view that if people choose to keep dogs then they must act responsibly and not burden others with them. I a not sure that "high-flying exec" is the right person to be taking on the responsibility of a dog! - but presumably she earns enough for the dog to go to kennels for 24 hours.
Our children do suddenly load the grandchildren on to us, but they know that we like this; but if they had a dog, they would not include this in the package. Maybe you could say you are happy to have the child but she will have to make other arrangements for the dog - or would this cause alarm and despondency out of proportion to the problem itself?!
To be honest, I do not think you are being a grumpy old lady - I think that I would feel used. It is your last sentence that concerns me. They are happy to load a job onto you when they are in difficulty, but cannot be bothered to engineer things so that you have company at Christmas.
We usually get round this problem by having the other sets of grandparents here, or we go there, so that everyone is included. But maybe you like the idea of a peaceful Christmas and it has just started to feel unreasonable when you have found yourself with a task that you are not entirely happy with.
What a difficult one for you. I do hope you find a way to resolve it - or that you just feel a bit better for having had the chance to share it with all of us.
Good luck and Happy Christmas!