Hello Shaked.
Sorry for jumping in…
I only watched the beginning of the video, and it was important to me to emphasize that Feiglin is not in favor of public transportation on the Sabbath. Feiglin supports privatizing public transportation, so that even during the week all transportation would be private, and for that reason he supports allowing public transportation on the Sabbath when it is operated by private companies. All of the Rabbi’s claims relate to transportation under the control of state institutions, and to the religious character of the formal state.
But it is not appropriate to force a private company not to operate mass transportation.
As for all state institutions, Feiglin argues that they should keep Jewish law, and therefore if the state does operate public transportation, it would be forbidden for it to operate it on the Sabbath.