Why does the Gemara use the lashon of "he'id" (101a) to say that "davar be'ito mah tov" refers to saying a pasuk bizmano?
We did not find any answer to this question in any of the books that discuss the rules of syntax used in Shas.
We did find, however, that the Gemara in Berachos (9b) says that Rebbi Yosi ben Elyakim was "He'id" that a person who is "Somech Ge'ulah l'Tefilah" is guaranteed that he will suffer no harm that day. The format of a statement of a Tana starting with "He'id" is the format of statements found in Maseches Eduyos, where the Mishnah collects various statements of statements that were said on the day that Rebbi Elazar ben Azaryah opened the doors of the Beis Midrash, letting everyone enter, as described later in Berachos (28a).
There are, indeed, a number of Agadic Mishnayos in Eduyos (see 2:9-10). Although the main point of the quote here in Sanhedrin and the quote in Berachos (9b) are Agadic in nature, they nevertheless are related to Halachic matters. The Gemara in Berachos is encouraging people to fulfil the Halachah of being Somech Ge'ulah l'Tefilah. The Gemara here in Sanhedrin is encouraging people to recite Pesukim at their proper time (that is, to learn about Pesach during Pesach, etc.; see MAHARSHA). Neither of these are obligatory Halachos (being Somech Ge'ulah l'Tefilah is a recommended practice, but not obligatory, and learning the verses about each festival is also not obligatory -- it is incumbent upon the community but not on the individual).
Perhaps these two particular statements of Agadah were stated on the day that Rebbi Elazar ben Azaryah was appointed because of their pertinence to the circumstances of his appointment. Raban Gamliel chastised Rebbi Yehoshua for ruling that Ma'ariv is only voluntary, in disagreement with Raban Gamliel's own opinion that it is obligatory. As a result of that, the people removed Raban Gamliel from the position of Nasi. The statement in Berachos was meant to show that even though Tefilas Ma'ariv is voluntary, one should still not take Ma'ariv lightly, because when one Davens Shemoneh Esreh right after Keri'as Shema of the evening, he is Somech Ge'ulah l'Tefilah (as the Gemara says in Berachos 4b) which has tremendous protective powers.
Our Gemara teaches of the importance of saying a verse at the proper time. Even if Ma'ariv is not obligatory, saying Shemoneh Esreh of Ma'ariv is certainly a proper thing at its time, just like reading verses in their time is proper, because this is the time at which Yakov Davened his Tefilah, and the time at which the limbs and fats are burned upon the Mizbe'ach (Berachos 26b).
The Agadic Mishnah in Eduyos is the one in which Rebbi Akiva teaches that a son inherits certain important qualities from his father. He may have quoted that teaching when he was offered the position of Nasi on the day that Raban Gamliel was removed, and he declined because he did not have Zechus Avos, as the Gemara in Berachos relates.