Since most of the major altcoins move up or down at the same time as bitcoin, aren't they all controlled? How does it work?

Pretty much every altcoin has a trade pairing with Bitcoin. Most of them have trade pairings with Ethereum too. If you want to buy an altcoin on a centralised exchange then you will buy them with USD, bitcoin or Ethereum most likely.
It’s both a blessing & a disguise I guess. When bitcoin pumps most alts tend to pump with a lag. Lots of money can be made in alts, although many of them have no use case or no real future, you can use Bitcoin to get in & out of them with nice profits.