Why does the institute make synths?
Solution 1:
This might contain spoilers, be aware:
I do not think Gen 3 Synths need food and sleep. While they look and behave like humans, those are still machines. They probably CAN eat, but they don't have to.
Human resources are limited and have to be trained. Most of the raiders where just reckless and brutal, not capable of solving complex problems. Synths can do that.
While you say humans aren't as loyal as synths, synths still are more loyal and can be developed further. You can tune a machine until its perfect, which was the goal of the institute.
Solution 2:
Without going into spoilers, when you meet a certain character, he explains:
... Scientific curiosity, and the goal of perfection, drove them ever onward. What they wanted was... the perfect machine. So they followed the best example thus far - the human being. Walking, talking, fully articulate... Capable of anything.
So it isn't to be slaves or to carry out cheap labour - It's just out of simple scientific curiosity and basically playing God.
Solution 3:
Humans require years of development before they can be used for any extensive labour, and it is easier to rationalize owning someone when they are not human (that is not possessing your sapience) and can be programmed for a particular task, not to mention humans sort of... resist being captured.
Also,
I suspect synths were further developed to secretly maintain contact with the surface after distrust from the surface folk forced them underground.
Solution 4:
History has shown that every time a society gets to use an underclass of people to take care of menial labor (and if managed well), the standard of living increases for its existing members. This was the primary motivation to use slaves, servants and these days, immigrants from poor third world countries.
The same applies to creating synths. By using synths, the institute hopes to drastically boost the standard of living for all humans that exist in that post-apocalyptic world. According to them, humans will no longer need to scavenge or hunt for food but instead, be left to pursue other interests such as creating art or advancing science while the Institute synths take care of the basics for humans.
Having established this premise, here are the answers to your questions.
Q. Why not use people instead of synths?
A. There are not enough people in the post apocalyptic wasteland. Too many of them are currently preoccupied with menial tasks such as producing food, scavenging or protecting establishments. When synths are created and used for these tasks, these humans will be freed up for "higher tasks".
Q. But there's no point using synths, since they will rebel anyway. Humans still make a good underclass
A. It's much easier to control a synth as long as you have its reset code. Just utter the recall code and the synth simply shuts down and goes into a sleep-like state. You cannot control a human that easily. This essentially reduces the cost of controlling a synth. One of the primary motivations to abolishing slavery, is that controlling humans is very costly.