From a regulatory standpoint, the threshold is not really "once airborne" at which you can use the published landing minimums, but when you actually divert to the alternate. At that point, it becomes your destination, so published minimums can be used. Before that point, the alternate forecast must remain within the criteria for use as an alternate.
From a probability standpoint, I believe the idea is to account for forecast errors within a pretty wide range of time, i.e., takeoff, en route, arrival, vectoring, miss, holding, decision, etc., while always ensuring a higher than normal probability that a legal landing can be made at the alternate, since it is assumed that no alternative beyond that exists. Once you commit to it as your destination, the time frame narrows considerably (hopefully, although in some parts of the world this is another discussion!), and the probability of a legal landing being precluded by a weather change diminishes quite a bit.