Which current constitutional monarchy gives the monarch power to the same degree that it gives power to the elected head-of-state?
I don't know a single one, maybe Thailand comes close? Though with > 50% of their senate members appointed by the military I would not even call them a democracy, not even close.
There was a popular candidate a party wanted as prime minister, who was not yet allowed to run as senator because of a sodomy conviction. So they tried to get his wife elected into a prominent position to streamline his path to prime minister. The party held the richest state in the country by a large majority and forced the head of state to step down, triggering a by-election. The wife won elections easily in that territory. It was a dick move because she is never physically in the area, but the area also didn't want to elect the Islamist party and voted for this more secular party. So she was voted as senator representing citizens in a region she didn't particular care about.
But senator wasn't enough, they wanted her in a head of state position. The sultan called out this BS and requested three names for candidates. The party submitted only one name. Sultan insisted on three names, and when they submitted three, he picked the second name as head of state.
So while kings don't have the power to block a democratic process, in this case, it prevented nepotism, which would also have messed up democracy.
In 2022, none of the three major coalitions won enough seats to form the government. Votes were split 38%, 30%, 22%. They all hated each other and part of the campaign promises were to bring down the other coalitions for corruption. The monarch ended up combining the parties into a unity government, which also entailed picking the Prime Minister.
I don't know a single one, maybe Thailand comes close? Though with > 50% of their senate members appointed by the military I would not even call them a democracy, not even close.