My best example for this question is a video Melevs Reef did a couple of years ago about the 20,000 gallon long island reef exhibit that has no controller. Of course, they have a full time employee who's job it is to monitor everything. Most of us have work and other things in our life.
A controller doesn't do anything you can't do yourself, but it automates things and does things for you when you're not home, making life significantly easier. There are plenty of successful reefers who don't use controllers, and plenty of reefers who say they'd never run a tank without one. Most of the people who don't have a controller still do use other 'controllers' - lighting timers, auto topoffs, heater controllers, etc.
One of the nice things I've found with a controller is that it lets you see things that you wouldn't otherwise see and adjust accordingly - I changed my alkalinity dosing schedule to minimize pH swings and realized my tank was getting hotter than I thought it was during the day.
Another advantage of a controller is the ability to do remote monitoring - when I'm out of town and have someone watching the tank for me, I can log in remotely and check on tank parameters. Things like flood sensors can also alert you to a malfunction ATO or float sensor to the skimmer reservoir before major damage is done.
One down side to be aware of is that it's very possible to program a controller incorrectly and cause major damage that way as well.
So basically, a controller isn't required, but it makes life easier. Even if you don't get a controller, Most people would still recommend getting an auto topoff system and a backup heater controller.
