Grany
Without a written constitution it's impossible to entrench anything in British law. Anything Paliament does Parliament can undo. If Labour introduces new rights in next Parliament, a future government can take them away. So written constitution and elected Head of State.
Something or someone has to be sovereign, Grany. It is either Parliament, elected representatives of the citizens of the country, or it is a Dictator (your 'elected head of state'). Which is going to best represent the diverse needs and desires of the electorate?
I prefer to think that Parliament, imperfect though it is, is the correct sovereign element. And a parliament without the power to change the law is not an effective instrument of governance.
I'm sorry, but I think those ideas, if they reflect the thought of Republicans, are pie in the sky. And dangerous. Systems which are subject to sudden change are volatile and vulnerable.
Written constitutions guarantee nothing at all. Even they can be changed and amended.