Current nodes do not allow replacebyfee unless it has optin enabled in the transaction that is going to be replaced. If not, you will have to wait till the transaction gets dropped (ie. Not being able to search for that transaction on blockexplorers) before you can broadcast it. It can take several days if no one rebroadcasts that transaction.
The thing is, I'm waiting for weeks now for the tx to be dropped. It just isn't. I have the feeling that the mempools keep the unconfirmed tx, to keep the fees high. Since the more tx are in the mempool, the higher fees you've got to pay to be confirmed. The whole bitcoin blockchain is messed up in my opinion
Not really. The mempool isn't managed by a single entity. Every node and miner has the rights to include or exclude any transaction in their mempool. The number of transaction will not affect the fees but rather, the fees paid in the transactions will do. In most cases, node's mempool will take your transactions out after 3 days or so since it takes up precious resources.
The rebroadcast can be done by anyone really. It is possible that the person you are sending the payment to is rebroadcasting it continually.
Anyways, if you really prefer not to wait forever, use Abdussamad's guide. CPFP can work well although it will take significantly more fees but it will get the job done.