Anything is possible.
Actually, no, some things are impossible. That is simply objective fact, despite the militant optimism characteristic in the USA.
The point that you are American and therefore would not do a good job as Queen Elizabeth is shaky. I would hope, given that today we see things like Idris Elba being talked about for the James Bond role, that there is room for inclusiveness.
No one is saying she couldn't do a good job. We're saying that no one in the UK will be able to hire her unless they can't find anyone else in the UK or EU who could do the job, and that contingency seems pretty remote. Also, I'm not sure why your example about a British man being considered for a British role is applicable. You know Idris Elba is British, right?
Tangetially, it really winds me up when people say "Anything is possible." That implies that if someone is unable to do something, it is that person's fault and not simply the result of circumstances. Some things are impossible, and people need to recognise that. No one is saying the OP shouldn't try, but if she does she should be prepared for failure that is not her fault. It may be that what she wants to do isn't possible, and telling her it is is unhelpful, and I think actually cruel if it creates false hope or makes her feel bad because she can't manage to make it happen. People need to have realistic expectations, whatever their aspirations may be.
/rant
ETA:
Perhaps.
But I would throw a flag if a person went in to audition for a role, nailed it - accent included - and then was told they couldn't have the job just because they were an immigrant. That would be wrong.
If they were already legally in the UK then you would have a point. But this is a question of sponsoring a visa. Apples and oranges.