The House of Lords serves as a check and balance against a government running amok.

But checks and balances from a body that by design is vastly conservative and somewhat religious is not a FAIR checks and balances.
If the government is elected democratically instead of first past the post like in UK, the checks and balances is democracy itself, but also the supreme court, as laws must align with the constitution.
Parlament is also a form of checks and balances.

So no House of lords is not a form of Checks and Balances, they are a form of oppressing the will of the people, so they don’t take too much power or money away from the rich. That’s what it was designed for, not as an instrument to improve democracy.

Ideally though, we could do with a House of … whatever’s below Common, because if the ones in the Commons are commoners, what does that make the rest of us?

Rulers will probably never be actually average. Even in a pretty good democracy. But I can say for sure, we are closer here in Denmark than the UK, because our democracy is better designed and more democratic.