Searching for this, all I can find is https://en.wikipedia.org/wiki/Rockoon where the balloon is lost with each launch.

What's the catch? Is such a cable not feasible? A 35km Zylon cable would weigh 50t to only support its own weight. Increase that cable weight budget by x20 to 1kt and it should still be little compared to the above example of 50kt of lift capacity. Zylon is commercially available at about $200/kg. For this project that would cost $200M. For a reusable rocket launch platform, that's still peanuts, right?