R. Huna said: If a man goes into a synagogue and finds the congregation saying the Tfillah, if he can commence and finish before the reader
reaches Modim, he may say the Tefillah, but otherwise he should not say it. R. Joshua b. Levi says: If he can commence and finish before the reader reaches the Kedusha, he should say the Tefillah, but otherwise he should not say it.
What is the ground of their difference? One authority [Rav Huna] held that a man praying by himself does say the Kedusha, while the other [Rav Yehoshua ben Levi] holds that he does not.
Rashi [ד"ה רב הונא] explains that according to Rav Huna a man davening by himself can say Kedusha so it doesn't matter if he doesn't finish in time to say kedusha with everyone else. However [here Rashi surprises] if he doesn't finish in time for Modim he still may say it on his own. The problem is that if he is still saying shmone esrei when everyone else says Modim he will not bow with everyone else and it appears as if he is denying haShem his due honor.
So we see that the issue is not the "Modim Dirabbanan" [the Modim that the tzibbur says when the chazan says his Modim] but the bowing.
Tosfos [ד"ה עד] agrees with Rashi and adds that the problem can't be that he will miss Modim Dirabanan because there is no real source for this tfilla in the gemara.
The Maharsha can simply not believe his eyes. He points to an explicit gemara in Sotah [40a] that mandates that we must say Modim Dirabanan, so how can Tosfos say that there is no source for it in the gemara??
[The Tzlach answers that the gemara means that the people said Modim Dirababan but it wasn't instituted by chazal as is borne out by the language of the gemara העם מה הם אומרים].
My teacher Rabbi Yaakov Homnick Shlita [Marbeh Bracha Simman 48] posits that Rashi and Tosfos learn that there are two separate reasons why we bow at modim. 1] We bow as part of the tfilla as we do for the bracha of avos. There we don't find that the tzibbur bows with the chazan. 2] A SPECIAL bowing that obligates the tzibbur which is not merely a detail of hilchos tfilla [as the bowing during avos is]. The bowing we do during Aleinu in Rosh Hashana and Yom Kippur is an extension of and under the same rubric as this special halacha.
Now we can answer the Maharsha's question on Tosfos [how can they say that Modim Dirabanan has no source in the gemara when we have an explicit one in Sotah]. Tosfos waas well aware of the gemara in Sotah. What Tosfos means is that we have no source in Shas that there is a SPECIAL, STRINGENT law from Chazal that one must say Modim together with bowing that would obligate one to finish shmone esrei before the chazan reaches that point. The gemara in Sotah implies that Modim Dirabanan was said as a regular tfilla but not a special obligation to bow and say the tfilla at the same time which would classify it together with Kedusha in its level of stringency. [It seems that this answer is along the same lines as the Tzlach we mentioned in brackets, namely that it was the people who were strict about this tfilla and bowing but not chazal. Or maybe he means that Chazal decreed the bowing as a detail in hilchos tfilla and it was the people who elevated the bowing to its special status. See there for more].
