Yes, a PID controller does not know of the system it is controlling. It is perfectly fine to turn the control signal to binary on/off that is beyond the scope of the controller itself.
What the system does is unknown to the PID. There might be some lag, some predictive component or anything in the system between it and the controller. So even though its a PID controller does not mean it has to controll what you think is obvious. Hell, the PID may be inside or control another controller**.
Second, the description of PID allows for many implementations to occur. The formula above is more a abstraction instrument, to categorize rather than a this is how you implement a PID*. A lot of things are PID controllers even though they may do the terms differently than the formula says. For example the integrative term may just be a history dependent term, nowhere does it say it has to integrate the error over the lifespan of the machine. Instead it may be a time limited slice too implemented like say a running average. The values may be capped and have some of the later systems features reduced into them.
* Although, since you can abstract to this then you might as well use it as a guide to implement. But then many things are PID controllers implicitly like spring/shock absorber systems and hydraulic pistons.
** So while one can be tuning a PID, does not mean the entire overall controller is a PID