I hate to rain on your parade.... but this is just plain silly.
Daily (diurnal) variations in air pressure will cause you to be forever adjusting for that "just balanced" point. Air pressure changes as different air masses come through will be even more pronounced.
Wind will have a nearly horizontal vector that will pull the balloons sideways and completely override any vertical component.
The "lifting" ability of the balloons is quite small, so the amount of power will be little, and stacking enough balloons to overcome that problem will result in such a large balloon mass that any movement will be very slow (meaning less output from your linear generator)
Of course, Helium is not an inexpensive gas.... and it escapes from weather balloons at an alarming rate. You'll be pulling them down to re-inflate them probably daily.
At best, you might make a few watts......but I doubt it would even cover the cost of the gas, let alone make any useful power.
There's also the issue of neighbours complaining, councils demanding permits, airports getting all upset about the balloons (or what might happen if they get away and foul commercial aircraft engines!)
The benefits of having everything on the ground are valid though
Keep thinking..... you might turn up something workable!