The OP is tendentious when he suggests that indirect election of the leader is not a “true” democratic system. It is not the American system with which he is familiar, but that is not the measure. I am sure the OP did not mean to be disparaging, but this is a field in which familiarity bias looms large.
Checks and balances are, of course, a good thing. But it does not follow that if some checks and balances are good, more must be better. What you can get with a parliamentary system that has an upper chamber that is too powerful is paralysis.
It becomes too hard to get legislation through. Real, necessary reform (on contentious issues like tax, migration, etc) becomes impossible, and so all that happens is endless posture-driven tinkering at the margins until the situation becomes catastrophic, followed by a potentially dangerous release of the tectonic political energies that have been building up over time.
And the method of election of the upper chamber (if that is to happen to the House of Lords) is very important. The LibDems are howling for a “fairer” system of election. They complain that the present one is “undemocratic”.
By having members represent individual constituencies, there is phenomenon known as “winner’s bonus”, where the winner’s proportional share of seats in the House of Commons is greater than their share of the primary vote.
That this should be so is obvious. If all seats uniformly voted 51% in favour of the Monster Raving Loony Party (for example) then the MRLP would get 100% of the seats. This is seen by the Lib Dems as “undemocratic”. In truth, however, the numbers never work like that extreme example, and the winner’s bonus is real but not overwhelming.
To my mind, this is a feature and not a bug. It allows for stable government. Most modern two-party dominated countries these days have organised themselves so that the dominant parties only ever get a margin a few points either side of 50%. If that weak majority was translated into seats in a House in a parliamentary system, there would be endless collapses, etc.
You can’t unscramble the egg once you have decided to change the system, and it is too easy to wind up like one of those disfunctional democracies whose well-intentioned but ill-chosen constitutional arrangements make stability impossible and whose goverments are a mish-mash of roiling and squabbling minor parties.
I am not British, so my interest in this is from afar, but for mine, the Lib Dems are being utterly self-serving by promoting the “fairer” proportional representation system they advocate for. That system too typically results in minor parties getting a crashingly disproportionate share of power, by holding the balance of power.
Thus, if Labour and the Tories each get 47 and 48% of the vote, then neither has an outright win, and a party with 5% gets to determine what happens.
That is far more undemocratic (and unstable) than the system which presently obtains.
This is why the LibDems want electoral reform - it is the road to power that they could otherwise only dream about, since they will never get elected in their own right. And why voting for the House of Lords (or whatever the new upper chamber is to be called) has to be considered with great care.
The Australian experience is that very often people vote one way for the Reps and another way for the Senate as a power check. The Reps are elected by seats, so there is almost no minor party representation there, but the Senate by proportional representation. The result has been the dominance of single issue party and minor party balance of power politics in the Senate. Why should some isolated and unrepresentative figure from Tasmania (or the Outer Hebrides, or wherever) get to have a power of veto over the government’s program, as has happened?
Why should that person (or tiny party) get the power to negotiate with the goverment on getting the government’s program through from a position of ridiculous strength amounting to blackmail?
What is or is not “undemocratic” is not necessarily obvious. There is not necessarily a linear scale for these things.