This doesn't really help me with anything, and I still can't replace her in any way. Replacing local MPs is hard too, since (at least my country) doesn't have a legal basis for a referendum to remove the current government and hold new elections, but locally we at least have pressure tactics in the form of posters, grafitti, protests, granite road tiles and guillotines.

british prime minsters are chosen as the head of the party, by members of the party. This thread seems to not understand that members of the british public can vote for the leader of the party they wish to vote for.

We do not vote on individuals, but on a group at election time. That group is democratically elected, and within that group the leader is democratically elected by its party members (whoever wishes to join).

The EU by definition is anti-democractic (any centralization of power without accountability results in loss of democractic influence). The most important thing to understand is that the public do not have the same level of influence on the direction of the EU that say the public has in in the UK. With that said, the position of the UK and its dealings is irrelevant to the matter at hand. If we are are discussing an anti-democractic group we don't make it democractic by finding someone equally or more so anti-democratic.

The British Prime Minister needs to be an elected MP. She would need to be elected and then "promoted" for it to be an analogy.

They are not just "appointed", all the elected MPs get together and have a popularity contest as to who is going to represent them. That's all the PM does, they are not the head of state.

She is closer to the House of Lords, where people can suggest bills, and no one votes for them and even if they decide to be evil, it's virtually impossible for the people to remove them.

The House of Lords is also a terrible institution that is designed to remove power from those who vote.