I think you've more or less got the right of it Ogaju. From what I understand (of the vast amounts of rumours and talk swirling around), White wants to have his personal doctors make decisions as to his ability to play, travel etc. The way some people were phrasing it was as his doctors, rather than Rockets' medical staff or front office, having veto power over decisions which may affect White's health. If that is the case, I think we can all see why the Rockets will never (and should never) sign this proposed addendum.
As you said Ogaju, I doubt the Rockets could or would stop White from seeing his own personal doctors, but I also doubt they will let them have significant power over basketball related decisions which "may" affect White's health negatively.
But then, that may all also be rumour! Unfortunately, we really don't know a great deal about thr situation as of yet. Inevitably, the details will arise and our curiosity will be satisfied, but I doubt it will be in the very near future. Until then, we'll all keep speculating :p