They were not introducing additional water into the system, they were saving and reusing the water which is naturally occuring.

Unless you would close the whole system, there is always a lot of water lost throught evaporation. Yes you can reduce that amount by covering the ground with plants, but the evaporation rate still is significant.
The naturally occuring water you refer to here is (in the dessert) almost only the ground water. Ground water is a big suplly of water, but esecially in the dessert it takes a long time to refill(you can observe this problem in spain. In spain the agriculture uses wells as water supply, but in the last years more and more wells get dry → the water reservoirs are not able to supply agriculture for an extended period).

Algae occurs naturally

Before we came there was an equillibrium between CO₂-producers and -consumers. Yes the equillibrium reacts(more algae) to the change we made with our emmissions, but the new equillibrium will have an overall higher CO₂-concentration, because there is more CO₂ in the system(more algae produce more biomass → biomass is consumed by animals releasing CO₂ back into the air → no high enough reduction of CO₂).

releases it slowly (most often in the form of methane).

methane is an even worse greenhouse gas than CO₂ → Algae make the situatio worse!