I think that I understand your point on re-compressing, but surely it's not necessary to take it back to liquid? Just more dense? Or am I, as often happens, being even more dense than Water?
I also note your point on maintaining purity, but again, I can't see why that's a major problem. Release compressed gas to empty balloons, then suck it in again, no other gas involved. Even allowing for leakage, I can't grasp why this would be anything other than Helium loss?
What I really need to have explained to me (and I've spent quite a while on Wiki) is why you run out of Helium buoyancy at relatively low altitude.
Warm the helium?
So, assuming that I've missed a fundamental property of Helium, why not go back to good old Hydrogen. More than quite a lot available. Does that avoid the property of Helium that I've missed?
And given that the Hindenburg crash was not directly caused by the gas (but as I recall) the cellulose-covered skin (dope) and that modern inerting and monitoring could raise the safety level several fold. Why not re-educate the public? Hydrogen is a touch flammable, but it's not that bad. Apart from wholly electrically powered aircraft (which will mostly be Lithium Polymer equipped (with it's attendant propensity to do hot stuff) I can't think of any 'planes that aren't a flying bomb in the wrong circumstances.
Oh. Gliders. Forgot them.
It seems that beer needs to be mentioned. Written after three NTP Guinnesses.