I understand how Bitcoin mining requires a long effort to guess the nounce until one is able to produce hash with leading zeros.
I have two particular questions here --
Why is Bitcoin mining made so computationally expensive in the first place? If the purpose is to just choose a random winner for block placement, why not use a simple and faster proof-of-work algorithm? (one example could be to generate a random number between 0-1 and the one with the smallest/largest value wins the round). By making the puzzle less computationally expensive, we should save lot of electric energy globally.
Is there any specific advantage of choosing a puzzle to produce resulting hash with leading zeros?
The difficulty of the algorithm is precisely what makes it difficult to cheat/steal on the Bitcoin network. If the algorithm was easy, then anyone could recreate old blocks and delete old spends so it looks like they never spent any Bitcoin after they buy something, for example. The purpose is not to pick a random winner, the purpose is to reward the miners doing the most work. It's true the winner is random, but the more work you do (more hashpower) the higher the probability that you will win. The probability is equal to the proportion of hashpower you're spending in relation to the total hashpower of the network.
Leading zeros is not what makes the hash valid, it just has to be under a threshold value. Leading zeros just happens because the number is low. It's like writing 1000 or 001000, it's still the same number, but the hash is 32 bytes so the leading zeros are there so you can see all 32 bytes.
I highly recommend reading the Bitcoin Whitepaper on proof-of-work. Also check out the Bitcoin Wiki - PoW