Pretty much all modern MoBos have built in hard disk controllers (The IDE ports). No they are not necessary for a computer to run, They are only needed if you want a hard dirve. I am sure that you could set up dual floppy DOS system if you really wanted to. It may be hard to find software which would run on it (forget about Windows). My first PCs (in '80 a Apple II+, then 8086 PC in '86) were both dual floppy systems, no hard dirve. The application disk went into drive A and a Data disk in drive B. YOu could even run your system with no foppys, but why?
Basically a HDD controller card unloads the task of controlling HDD's from the system board. Most controller cards have faster controllers than those integrated into less expensive system boards, some controller cards also allow for more than two IDE channels. Quite a few IDE controller cards also include a parallel port and/or a serial or gameport. Yes, most (all?) new system boards integrate an IDE or SCSI controller, and so a controller card is not necessary for newer computers. So, technically, a "controller" is necessary, but a "controller card" is not, because most system boards have the controller built in and do not need an add-in card to do that work.