This is a classic case of a practice that has changed from the manner it was originally intended, with splintered variations arising. Let us proceed through the development.
According to many authorities, originally, a mezamen would not only introduce bentching with what we call zimun, but would also recite all of Birkat HaMazon aloud, while the others would listen and answer amen.[2] This accomplishes most fully the idea of praising HaShem together.[3] The minhag has developed over time for every individual to bentch by himself, apparently out of concern that people will not listen to the mezamen well enough[4] or because they may have to understand the text they are hearing in order to be included.[5]
What, if anything, is left of the idea of a joint bentching? When the Shulchan Aruch[6] says that every person bentches by himself, he notes that they do so quietly. In this way, they can still hear the mezamen.[7] The Rama[8] adds that the others should go ahead toward the end of each beracha to enable them to answer amen to the mezamen’s berachot.[9] Many people follow the Rama’s suggestion (except that the mezamen usually waits for the others, instead of the others speeding up).
The Mishna Berura[10] mentions that in his time it was common for everyone to bentch out loud, and no one heard the mezamen. (Now it is more common for everyone, including the mezamen, to bentch quietly.) However, the Mishna Berura stresses that it is important for all to hear the mezamen at least for the first beracha (until “hazan et hakol”) because of the halacha[11] that this marks the end of the zimun. (The main ramification of this idea is that those who interrupt their meals to answer to the zimun should wait until after that point before resuming their meals.[12])
The matter depends on a machloket Amoraim[13] regarding whether zimun ends at “hazan et hakol” or at “u’vetuvo chayinu,” the addition to bentching that is inserted when there is a zimun. Sephardim follow the latter opinion.[14] The Mishna Berura,[15] ruling for Ashkenazim, instructs that one should read along the first beracha quietly at the pace of the mezamen, who should be reciting it out loud.[16] The Magen Avraham[17] goes a step further, saying that until that point, people should only listen to the mezamen and then bentch the remainder by themselves. The Mishna Berura[18] writes that only those who can concentrate on listening to the mazamen and can understand the first beracha should follow the Magen Avraham.
As you observed, the mezamen does not always recite even the first beracha for the others to listen. Because it is difficult to quarrel with a prevalent practice that has been followed by knowledgeable people for a long time,[19] different rationales for the leniency have been given. One is that, in this case, Ashkenazim rely on the Shulchan Aruch’s ruling that zimun ends with “u’vituvo chayinu.”[20] In fact, the Tzitz Eliezer[21] cites an opinion that it is generally better to bentch separately after a zimun, without listening to the mezamen. Additionally, the Chazon Ish[22] indicates that when there is a zimun of ten, in which case HaShem’s name is invoked in the introductory part of the zimun, it is not necessary to listen to the mezamen until “hazan et hakol.”[23]
In summary, in places where there is no clear minhag to the contrary, we recommend that one follow the Mishna Berura’s position. Namely, the mezamen should recite the first beracha audibly and the others should read along quietly while also listening. However, we do not discredit the other approaches you have seen.