It's probably inside for mechanical reasons pumping liquid helium is a nightmare it sips through pretty much everything (it can even go through glass when sufficiently cooled), in this case you don't really need to worry about all that stuff since you simply let the helium out and the difference in density would sort everything.
As for the reason it's not used to "cool the helium" it's because helium boils at -268.9c the liquid oxygen is cooled to -182.9c or to ~-206c in cryogenic mode, this is still way above the boiling temperature of helium. This would be about as effective at cooling, as cooling boiling oil with boiling water, incidentally it's also about as explosive ;)