There are many cases where you really really want a stay until the appeal is heard.
If a someone can get a win against their competitor in a lower court and convince the court of an injunction/massive fine (such as in a copyright or patent claim) then that may be enough to drive the competitor out of business.
The competitor may have no revenue coming in until the time the appeal(s) are all eventually heard, even if the appeal is ultimately successful (due to support from better lawyers/experts) it would be too late as the damage is done.