What makes it so much faster to write to the "WAL" compared to directly to the "real database"?
You must have missed the answer while reading documentation:

Using WAL results in a significantly reduced number of disk writes, because only the log file needs to be flushed to disk to guarantee that a transaction is committed, rather than every data file changed by the transaction. The log file is written sequentially, and so the cost of syncing the log is much less than the cost of flushing the data pages. This is especially true for servers handling many small transactions touching different parts of the data store. Furthermore, when the server is processing many small concurrent transactions, one fsync of the log file may suffice to commit many transactions.