Fair point, but we haven't had a democratically elected Prime Minister for quite a while now, let alone the House of Lords!
The issue is whether the leaders of the commons should be able to appoint people to the upper chamber for life, whether the church should have a seat at the table etc. Or whether the upper chamber should be directly elected. It’s a matter of governance and constitution more than embedded privilege.
So as much as “Guillotine the nobs” would be a popular cry, most of them aren’t nobs in the first place.
French Revolution - 1789.
English Revolution - 1649.
So the English cut off their Monarch's head before the French did. But we them invited the Monarch's back after not liking the republic of the "Lord Protector".
Admittedly the English state, not the later British state, but anyway...
If the UK really cared about unelected government, why don't they get rid of their House of Lords? This would have been much less painful than Brexit. It looks a lot like this unelected government complaint is just an excuse and not something that they actually care about.