Well...sort of. It's not really "home tuition", at least at Glasgow. EU residents get discounted price, UK even less(and depending on where--N Ireland, Wales, England I think pay different rates) , and Scots are free. I believe Scots at RVC have to pay?
From my experience, students who hold dual citizenship usually do not use the US one to apply, because even the second degree EU/UK cost is cheaper. But if you do that and need US loans, it's a no go.
I understand the rationale behind the second degree bit, but I hear your point as well. Free education is fantastic, but they need to get money somewhere. Vet degrees are Bachelor's here, so if you decide in secondary school you want to do English, that is what you are stuck with because you don't pay. If you change you mind and go back, well, they need to make money somehow and not have people play the system too much. All the schools are now realizing they are falling under though, which is why there is the hike in tuition for everyone(up to 9k). Students are not fans, but to keep the universities afloat, it makes sense. Plus, if they need loans, they don't have to pay them back until they are in steady work after graduation. Though I have to say, my undergrad still cost me more than this degree, even with the conversion rate.
Maybe RVC will recognize you new citizenship? I would ask. The worst they can say is no. Good luck! The UK is a fabulous place to be!