So I'm going to make an analogy here, not so much for you Alexander but also for new people reading this !
This situation is comparable to a movie theater. If i decided to put a sign on my house that says "watch movies here for $1" , and I'm showing the latest films at my house for $1 to the public, I can get arrested. The reason being that people took their time and talent to make that movie, and I'm showing it to people without the movie creators receiving what THEY deem a fair cut of the money. Movie theaters on the other hand ARE aloud to show movies because they have agreements with Disney, WB, etc.
It's the same thing for YouTube and using copyright music.
The TVnetworks, and Music Radio Companies are NOT the people you need to beg and message personally. (That would be like begging your local movie theater to allow you to sell tickets for people to watch StarWars in your house)
The people you would need to get in contact with would be the music publishers - you would need to get in content with Katy Perry and Drake - or the people in charge of them, then you would have to buy the rights to use their songs. They would then get a cut of the monitization money from your video and so would you ! This is similar to how a movie theater has a contact Disney to pay them large amounts of money to show one of their movies, when a tickets sell's - Disney gets some money and so does the theater.
So if you 100% must make this top ten video LEGALLY you will need to own a Radio Station / Music Network of some sort, and you will also need a VERY LARGE some of money to pay Drake, Ariana Grande, etc to use their songs. ( this is why spotify has so many ads - and this is why the radio station has ads too ! The song writers get paid this way which is also why downloading music off the internet is illegal - how is Drake getting paid if you're getting to listen to his song for free)
To answer your original question "how are some people monetizing famous songs" :
The people who do this legally have lots of money and probably own a radio station ! The people who do this illegally are just so happening to get away with it somehow - and eventually they will be caught.
so instead of
"how do I post famous songs on YouTube ?"
You need to ask is:
"What step do I need to take to become a music radio network thing so that I can post famous songs on YouTube"
And sadly Myself and many people probably can not answer that question.
I don't know how to start a radio station.
After the thousands of dollars you would have to potentially spend to make this YouTube video LEGALLY you must remember that it does not ensure your videos success.
After you go through all the work to become a music network someone like MTV could steal your idea and have way more success because they have thousands of subscribers already ):
However I can see your determined to do this Alexander and that's good, so I wish you good luck ! Ask around to more people and maybe get some other answers - I do have a pretty firm understanding on copyright but I belong more in the realm of gaming, as I am a gaming channel. Don't just take this advice from me. Maybe someone has another answer.