Wouldn’t this all be very similar to the concept of the electoral college? In theory the electoral college doesn’t have to vote for who the people vote for, but they know if they decided not to then all hell would break loose. In the same way the Queen has, in theory, power but she knows if she tried to execute it then she’d never get away with it and could possibly end up seeing the monarchy abolished.
A bicameral legislature is not democratic if one of the houses has no elected members.
Only if the unelected House has any actual power; which- as it has been well established- the House of Lords does not. Even in your earlier example the Legislation was still passed, despite their attempts to the contrary.
They ended up losing because there was a small concession and they ended up voting for the act, not because they were pushed out of the way.