As you seem to be aware, when there are ten men present to begin a part of tefilla and then someone leaves, the rest of the group can complete that unit of tefilla, as long as six men remain.[1] There is no indication in the poskim that this depends on whether all of the original ten had been davening with the minyan or that some were only responding. However, the Yerushalmi[2] states that it is a sin and a bad omen for the one whose departure is responsible for the lack of a minyan.
Several Acharonim discuss your question of whether the group can use the halacha of continuing when it knew from the start that it would lose the minyan.[3] The Rav Pe’alim[4] deduces from the Yerushalmi’s language (although one can argue) that a group must not start a unit of tefilla if it expects to lose its minyan before it reaches the unit’s end. The logic is that since being left in the middle of a unit without a minyan is a negative situation, one may not knowingly create it. The Pitchei Teshuva[5] is unsure. The Levushei Mordechai[6] rules that it is generally permissible.
A number of responsa deal with the matter from the perspective of a person who was asked to help complete a minyan but has only a few minutes to give. Should he put himself in the situation of leaving, and thus ostensibly be subject to a bad omen? The Ohel Yissachar[7] assumes that his leaving would be problematic, and he is therefore justified in refusing to join the minyan. The Shevet HaLevi[8] says that only if the “helper” has a very important reason to leave may he join the group and then leave in a way that they will no longer have a minyan. He posits that when one has justification to leave, the Yerushalmi’s criticism does not apply. This seems especially relevant in a case in which the one leaving was not davening as part of the minyan, but was only helping as much as he could.[9] Similarly, Rav S.Z. Auerbach is cited[10] as asserting that the ability to help create a minyan outweighs the negative elements of leaving it early.
Your case has a possibly lenient element that is absent from the cases that most of the Acharonim discuss – your helper stayed until the end of chazarat hashatz. The Mishna Berura writes that one is permitted to leave the minyan before the next unit is begun.[11] Although the end of chazarat hashatz might appear to be such a moment, the Terumat HaDeshen[12] and the Rama[13] write that even after losing a minyan in the middle of chazarat hashatz, the group can say the first two Kaddishes afterward, because they relate to chazarat hashatz. Thus, we see that the unit that includes chazarat hashatz does not end until after the second Kaddish that follows it. The Mishna Berura[14] writes that consequently, in order to avoid the Yerushalmi’s criticism, one should not leave the minyan before the second Kaddish. On the other hand, the Eshel Avraham[15] rules that leaving is forbidden only during chazarat hashatz, which consists of berachot that are justified only with a minyan, whereas reciting Kaddish without a minyan is less severe. Furthermore, logic dictates that concern for a Kaddish that will lack its full impact should not cause a group to miss out on a full chazarat hashatz, which is an older and stronger obligation.
We will conclude with an innovative idea that links stringency and leniency regarding your question. Since the helper is following the group’s request within his limitations, it is the group more than the helper that is responsible for the minyan disappearing in the middle. (In the classic case discussed by the poskim, one leaves against the others’ will.) Therefore, the fallout should be on the group, and to avoid it, the group should look for another “helper” to take the first one’s place (there is no need for the same ten men the whole time). Realizing the stringency of avoiding a disappearing minyan, they should make a strong effort, and it is therefore reasonable for the group to assume it will find a replacement. If it is not successful, the situation is like the standard one in which a group expected to have a minyan the whole time but then lost it. (Of course, this argument will not work if there is no real possibility of finding a replacement.) Therefore, it was certainly worthwhile for you to have made use of the tenth man while you had him, and, in the future, one should not be deterred from agreeing to stay for as long as he can.