I'm not sure closed-cycle coolers can get as cold as evaporation of liquid helium? Herschel was cooled to ~1.7 K [0] while MIRI will only be cooled to 7K [1]. Also, with regard to reliability, evaporation of liquid helium is a pretty foolproof design, while closed-cycle systems can potentially break and end a mission early.
[0] https://en.wikipedia.org/wiki/Herschel_Space_Observatory#Ins...