Usually it's one full ruling. And then separate concurring or dissenting rulings. ICC style.
Many courts operate that way (ICJ, European Court of Human Rights, SCOTUS), which is the best way IMO, and a middle way between two opposites: one is many individual pieces representing the majority and the other is where you have one piece remaining completely opaque about individual judges' thoughts (like the European Court of Justice). There are pros and cons to all but this one where you have one main judgment with separate opinions for those who wish to write them provides both clarity and some ability to "interrogate" individuals and study their judicial philosophy. Multiple ones without one of them being the "official" one would definitely lead to confusion where there are contradictions on finer but crucial points between judges who otherwise agree in large part.
Even if the decision is split between the judges, so some write on some issues and others different ones, as long as they do so on behalf of the majority and this is clear, it should be fine. Maybe NASA thought that asking for individual pieces would lead to each judge being more "responsible" but I think its always better to have a major decision at least on each of the points of dispute and then where judges feel they agree with the conclusion but not some aspect of the reasoning or they wish to stress something further as you put it, or just dissent completely, they can do so in separate opinions. But lower courts will be better served with an authoritative reasoning that represents the majority decision on each point of dispute so that they will know what the law is. In any case, the court is free to develop its own tradition in this regard like other top courts, parties can't force them.