The amount of transaction fee has no influence how fast the next block is found. Finding a valid blockhash is a random process, the more and faster miners try, the quicker they statistically find a valid blockhash.

To get a transaction in the very next block it needs to have a competitive enough fee (or more), so that every mining pool has a financial benefit to include your transaction, because your transaction competes with other transaction for block space.

Since quite some time there are usually more transactions pending to be mined in a block than fit in the next block, so naturally there is competition for block space. Miners normally choose transactions based on fee rate, starting with highest paying until the block space for transaction is used up (this is very simplified).