u/KinginJello answered this question years ago. Let me locate the quote…
I think a lot depends on the process by which these women distribute swords that then confer governing authority.
If we assume that by 'strange,' you mean that their motives and decision processes are opaque to us, sword-distribution-as-election has one significant benefit over democracy, especially democracies like what we have in the U.S. Here, becoming a major officeholder (think President, Senator, or Supreme Court judge) requires a lifelong, single-minded commitment to the pursuit of power. You have to raise your profile through smaller elections, you have to build your own fortune or raise a huge amount of money, you have to endure lots of humiliation, both in the form of press scrutiny and sucking up to people you don't like because of their influence or wealth.
What this leads to is an environment where only people who are truly power-hungry would ever end up in our most powerful governmental roles. And hunger for power often goes hand-in-hand with very undesirable traits for rulers.
Lake Sword-based autocracy, however, because of the 'strangeness' of the sword distributors, can't be gamed in the same way, and so results in something more like a lottery, where people are chosen for government irrespective of their desire to be powerful. They would also choose them irrespective of their qualifications, but I think, if you look at our last three presidents, you can find at least 50% of the country who thinks each of them was totally unqualified, so it's not like democracy is knocking it out of the park there.
Now, your question leaves the door open to only using the sword distribution as the 'basis' for the system of government, but not the end-all, be-all. You could set up a system of checks and balances, whereby the sword recipients pass and enforce laws, but those laws are able to be vetoed by a representitive body or even a plebicite.
We could make it work.