In my community (I am the rabbi), we daven Mincha during the week without a separate chazarat hashatz (=heiche Kedusha) because of people’s busy schedules. In the winter, we have the practice of davening Mincha of Shabbat after the shul Kiddush following Musaf. Some congregants have requested that we do short Mincha, as their wives wait to go home with them. Is there any basis to allow this? Answer:
Chazarat hashatz was instituted after the silent Shemoneh Esrei and for the purpose of providing Shemoneh Esrei for those who cannot daven themselves (Rosh Hashana 34b). Of course, we continue doing it even if no one needs such a service, and it has a special status of tefilla in and/or of the tzibbur. Chazal also instituted that the chazan recites a silent Shemoneh Esrei before chazarat hashatz, even though that could have fulfilled both his private and public obligations. It is done so the chazan can "practice" before chazarat hashatz (ibid.).There was a time when heiche Kedusha was done with the chazan continuing to recite the amida out loud while individuals were saying it quietly (see Radbaz IV:94; Magen Avraham 232:2). The way we do heiche Kedusha (the chazan stops reciting the amida out loud after HaKel Hakadosh), we miss all of these elements, and what is left is the ability to recite Kedusha together.
The Shulchan Aruch (Orach Chayim 124:1) allows one who needs to be chazan but comes to shul very late to recite chazarat hashatz "without practice." The Rama (ad loc.) notes that if they will not be able to recite Shemoneh Esrei and chazarat hashatz before the appointed time, the tzibbur may start Shemoneh Esrei with the chazan. (In Living the Halachic Process III, A-2, we discussed whether the tzibbur should start immediately or after Kedusha.) Another situation of need brought to justify heiche Kedusha is when it is unclear if the requisite number of people will answer amen to chazarat hashatz (see Radbaz ibid.). The Beit Yosef (OC 124) relates that the minhag in most congregations was to regularly do a shortened Mincha but he does not cite this as halacha in the Shulchan Aruch. The Darchei Moshe (ad loc. 3) reports that this was not the minhag in the communities he knew of and permits it only for cases of need. Nevertheless, a reasonable minority of congregations (like yours) always posit that they have enough need to shorten the davening at Mincha during the week, which is a local rabbi’s call.
You ask if this can be done on Shabbat, for a new need – so that wives do not have to wait too long for husbands. We have found opinions that restrict when one can do heiche Kedusha. The Pri Megadim (EA, OC 591:1) raises the problem of fulfilling one’s amida obligation with chazarat hashatz on a day that piyutim are said, as they can be a hefsek. Another questionable situation is on a fast day where the chazan is not able to recite Aneinu as a separate beracha (see Magen Avraham ibid.; Biur Halacha 232:1).
Of course these problems do not apply at a regular Shabbat Mincha, and we have not identified other problems. The practice of chazarat hashatz is not significantly different on Shabbat than during the week. We have not found sources that preclude heiche Kedusha. While there is little literature on the topic of doing so, the fact that a minority of Sephardi communities do so for Shabbat Musaf regularly, without special need, lends credence to its halachic legitimacy.
You are likely bothered by the lack of a minhag to do short Mincha in communities you have seen, which is a valid concern. However, this does not necessarily mean there is a minhag against it. Rather, on Shabbat it is rare for there not be enough time or that the minyan is so weak that this is necessary.
Thus, it is a question of advisability. To what extent is lowering the level of an element of tefilla justified to encourage more people to come (or stay)? To what extent does it foster harmonious relationships within the community and its families? You are more equipped to answer than we are.