While I might normally put these as comments, I think this can warrant an answer.
Firstly, it is possible that even with piercing just their sexual organs, they may have died from blood loss from the wounds that may have occurred.
Secondly, the Gemara (Sanhedrin 82b) which you quoted says the following:
א"ר יוחנן ששה נסים נעשו לו לפנחס אחד שהיה לו לזמרי לפרוש ולא פירש ואחד שהיה לו לדבר ולא דבר ואחד שכוון בזכרותו של איש ובנקבותה של אשה ואחד שלא נשמטו מן הרומח ואחד שבא מלאך והגביה את המשקוף ואחד שבא מלאך והשחית בעם
The bold parts loosely translated mean:
Rabbi Yochanan said: Six miracles were performed for Pinchas... ...And one that he was able to direct the spear into the male sexual organ and the female sexual organ, and one that they did not slip off of the spear. (my translation)
Thus, the fact that they were speared exactly at that location and died may be a possible interpretation of what the Gemara considers to be the miracle here.
Now I also found Targum Pseudo-Yonasan here, who suggests that they remained alive for quite a while, and only died after being paraded through the entire camp.
Edit: Hazoriz notes the Sifsei Chachamim/Mizrachi here and here, who say the same.