The Mishna, in Ta'anit 4:1-4, describes an institution called ma'amadot: Just as the Kohens and Levites were divided into 24 clans, which each handled the Temple service for two non-holiday weeks of the year, the non-Levite Israelites were also divided into 24 groups, called ma'amadot. These groups also rotated being on-duty through the year, but instead of serving in the Temple, members of the week's ma'amad would assume "ownership" responsibility on behalf of all of Israel for the public offerings brought in the Temple that week, particularly the Korban Tamid. The mishnayot and the Gemara on them (Ta'anit 27a-28a) describe practices the members of the week's ma'amad would undertake, including special prayers and Torah readings and fasting.
We are not bringing offerings in the Temple now, but we do still take responsibility for the daily Tamid by reading the Torah's instructions for it before prayers. The Talmud here explicitly calls out this substitution. In a Midrashic passage, God tells Abraham about the Korban Tamid, and then:
אמר לפניו רבש"ע תינח בזמן שבית המקדש קיים בזמן שאין בית המקדש קיים מה תהא עליהם אמר לו כבר תקנתי להם סדר קרבנות בזמן שקוראין בהן לפני מעלה אני עליהם כאילו הקריבום לפני ואני מוחל להם על כל עונותיהם
Abraham said before God: Master of the Universe, this works out well when the Temple is standing, but when the Temple is not standing, what will become of them? God said to him: I have already enacted for them the order of offerings. When they read them before Me, I will ascribe them credit as though they had sacrificed them before Me and I will pardon them for all their transgressions.
So, it seems to me that it could be valid and valuable to organize modern-day "ma'amadot," focused around the recitation of Temple service passages instead of the Temple service itself, and presumably omitting drastic changes in the established standard liturgy. I don't expect that I'm the first person in the last 2000 years to consider this. Has there ever been a prominent proposal or movement to recreate such an organization?
There is an extensive collection of readings toward this sort of purpose printed in some siddurim called Seder Ma'amadot. I'm asking about whether anyone has proposed a system of rotating focused responsibility.