Probably it was the second senator to do that -- John F. Kennedy -- who really started the flood, but prior to 1914 people recognized that being elected or serving as a senator wasn't especially good practice for being president. Afterwards, senatorial races were recognized as run-ups to presidential campaigns. And Senators earn their reputation from statesmanship or orator as Webster and Clay did, but from personal magnetism, vote-winning power, and getting their name on legislation that might be unnecessary or harmful.
And over time, it meant that senators came to look at things more from the point of view as potential presidents, rather than as local representatives. It didn't happen all at once, though. Through much of the 20th century, senators were still concerned mostly with local issues (segregation and military bases for Southerners, water projects for Westerners, housing and labor legislation for Easterners). Kennedy and TV had a lot to do with the change, but whether direct election was the cause of government becoming more centralized in Washington, or a result of it is something to argue about.
People are right in pointing out that direct election didn't do one important thing it was supposed to do -- eliminate corruption. But then they seem to argue that it created greater corruption and that going back would clean up the Senate. As you say, the 17th amendment transfered corruption from one sphere to another, and going back would likely simply shift it back to the state legislatures.