"Still fuck all, except a prime minister who nobody elected"

We-e-e-e-ll, sorry Don, he was elected. Elected as an MP by his constituency, and elected as leader of the party-in-government by their party election process. He underwent the full election process necessary to become Prime Minister, and his occupancy of that position is perfectly legitimate. Whether he's the right person for the job is open to debate, but his position as Prime Minister is unquestionably legitimate.

If we follow your logic, Tony Blair wasn't elected, Sir Winston Churchill wasn't elected, Lloyd George wasn't elected, even your heroine, The Thatcher Bitch, wasn't elected. It's not a presidential system here, the general public don't get to elect a Prime Minister - ever - just their own constituency MP. The Prime Minister is elected within his/her own party, so get over it.