the originals is the best show I ever watched I wish the season could continue without any interruption

And just to be clear – No, No and No.

Both “Supernatural” and “The Originals” share the story thread of being about two brothers,

Except, the brothers in Supernatural aren’t always trying to screw each-other over.

they each have a wide array of opponents with all sorts of powers.

Sure – a wide array. On Supernatural we’ve had gods, demons, angels, archangels, mutated angels, Leviathan, Mother of all monsters, Alpha monsters, witches, horsemen of Apocalypse – and these are just the major adversaries. Not to mention all the countless kinds of minor monsters they’ve faced.

And on Originals we have – witches, vampires, werewolves and…. no wait, that’s it. So, I guess not that wide an array after all.

…could have the Winchesters questioning what to do with [Klaus].

How is that even a question? Both Klaus and Elijah have killed countless innocent people in that past, they continue to kill people today and they turn others into monsters who keep kill humans as well. OTOH, killing the two of them would eliminate most of the vampire population of the world and save countless lives. Its a complete no-brainer – any crossover that does not end with the Winchester making it their mission to eliminate the Original family would be unfaithful to the Supernatural universe.

We also envision in our head a scene where Sam and Dean get Elijah to
open up by taking him to that karaoke bar where Demon Dean partied at
the start of season 10.

I can envision it too – with them getting Elijah to open up using a couple of machetes.

…it would be a nice reminder to Sam and Dean that while they are great at
taking care of problems in small towns, there are systemic examples of
supernatural creatures wrecking havoc in big cities. This hasn’t been
acknowledged much since “Bloodlines,” otherwise known as an episode most
people are trying to forget about.

There is a reason why people are trying to forget that episode – it went totally against everything Supernatural is about. And the reason for that is it was too much like Originals.

The idea that supernatural creatures wreak havoc in big cities in large organized groups without a whole lot of hunters taking notice is completely foreign to Supernatural. The Winchesters have had plenty of hunts in big cities and the principle remains the same – if a monster starts acting up, they take care of it before it can gather others of its kind. If the Originals’ New Orleans existed in the Supernatural universe, then there would’ve been a permanent contingent of hunters stationed there.

The case against a crossover:

1. Different mythologies (How does the Alpha Vamp square with the Original Family).
2. Logistics.
3. Both sides come off looking like idiots. How did Sam and Dean not notice the ridiculously high body count in New Orleans? And what about Klaus – who thinks that he is the oldest and most powerful supernatural creature living – when he faces off against the King of Hell or an Angel of the Lord or maybe Darkness herself?
4. One side would have to be OOC – either the Mikaelsons start caring about the collateral damage they cause or the Winchesters stop caring about innocent lives.
5. Different MO’s – everyone in the Originals is always trying to come up with convoluted and often ridiculous plots to defeat their enemies. With the Winchesters, the things are much simpler – figure out a way to kill it and then kill it.
6. Different dynamics – The MO on the originals is to manipulate and blackmail everybody – from family to friends to casual acquaintances. The Winchesters tend not to go along with that sort of thing.

So yeah, I’d say that a crossover between the two would be a pretty terrible thing for Supernatural. In fact, I think it’d be a lot like that terrible episode everyone is trying to forget.