Why are the CJ2+ anti-ice and de-ice systems separately controlled? Wouldn't it be easier for both systems to be controlled with one button/switch (minimize chance of error from occurring)? Are there any aircraft that have the two systems controlled by one button/switch that would be activated as soon as ice is detected?
Because they do two different things.
To quote FlightSafety's CJ2 Pilot Training Manual:
Anti-ice protection systems are incorporated into wing, engine components, windshield, pitot-static, and angle-of-attack systems, and pylon ram-air scoops. These systems should be activated prior to entering icing conditions.
The tail deice system consists of pneumatic boots on the horizontal stabilizers.
Basically, anti-ice is used to prevent ice from forming in the first place on the protected surfaces of the aircraft. This is done where it would either be "bad" to have it form in the first place (pitot-static, angle-of-attack, etc.), or where it would be "bad" to have the ice come off after it has formed (and then go through an engine, etc.).
Deice is used to remove ice after it has formed. In this case it is used on the tail which a) isn't critical if it has a little ice, and b) the ice can be broken off after it has formed and it will just fly off the back of the airplane without damaging anything.
Since they serve two different purposes (and protect different parts of the aircraft), there are two different controls/switches.
think of it from an MEL (Minimum Equipment List, manufacturer/governing body approved manual for working equipment & non-working equipment installed on aircraft) stand point if you have 1 switch controlling both systems and the fault lies within the switch, and you don't have a spare (or mx personnel) at the outstation with known-icing conditions probably means you aren't taking off. Yet if you separate systems you might be able to get airborne and play dodge cloud then, remaining clear of known icing.
As for specifics to that aircraft I am not privy as I don't fly it. Other aircraft that are controlled by logic and automated are pretty self explanatory if you understand the nature of its function.