I’ve seen some Modernist-sounding* texts that use the word “assembly” to avoid the word Church. Instead of addressing the congregation, the priest (er, “presider”) addresses the “assembly of the People of God.” Etc. In Hebrew and Greek, the word we translate as Church does in fact mean “assembly.” I surmise that the people who prefer “assembly” do so because they think the word has a more democratic connotation than the hierarchical “Church.” One thinks of the expression “popular assembly.” Implicit in such a usage is a critique of “liturgy as court ritual,” a critique that has been debunked elsewhere.**
Now, one of a Traditionalist or even just plain orthodox-with-a-small-o persuasion might object to this usage of “assembly.” He might advance any number of arguments, the premises of which we don’t share with the people who push for the “assembly” understanding of the Church, and he might cite a bunch of Church documents, originally written in Latin, that no one who has a job has time to read. In these scenarios, I look to my mentor in evangelism and apologetics, Vin Lewis of All Roads Ministry (https://www.allroadsministry.com/). Vin would say to use the simplest, most direct, most memorable argument that requires the least specialist knowledge (such as Latin, Greek, and Hebrew). So, with that in mind, here is how I would respond: You use the word “assembly” with the implication that this word suggests democracy, equality of all believers, lack of hierarchy, lack of formality and sharply defined roles at Mass, etc. However, the word should not imply that in modern American English, at least when we consider the two scenarios where we still have assemblies in our culture:
1.) School assemblies. I have been to dozens of school assemblies. Adults who were not answerable to me in any way ordered me from point A to point B to listen to them tell me things for my own good. I was not allowed to object, and I was punished if I did not attend. At the assembly, I might have a chance to speak, but only at a defined time, and I was allowed to speak only in the specified form (such as a question for the speaker) and on the topic set by the person who called the assembly. When the assembly ended, the adult in charge formally dismissed me.
2.) State assemblies. In the states where I have lived (Illinois, Wisconsin, New York), either the entire legislature or the lower house of the legislature is known as the Assembly (or General Assembly). The Assembly consists not of random citizens, but of select individuals, duly elected according to a constitution. If I as a citizen walked into the Assembly and started talking, or tried to set the agenda, or tried to interject in some way, I would be escorted out. The Assembly members are charged to speak on my behalf, and I am charged to remain silent, unless I am called on to testify. Sometimes the Assembly closes a session and I am kicked out. Not even the Assembly members get to speak at will, but all must observe the specified parliamentary procedure of their own state. Lastly, the Assembly itself has a defined constitutional role and can sometimes be overruled by the executive branch (the governor) or by the judicial branch (the courts). And they call this democracy!
Both examples sound a lot like a hierarchical liturgy where duly appointed clergymen summon the predominantly lay assembly. The members of the assembly are bound to speak only at designated times and in the designated form, such as the Gloria and Creed, and only in accordance with their specified offices. The priest or other preacher has a monopoly on speaking during the homily. At times, I as a full citizen of the Church am metaphorically “expelled” from the ritual, as when the priest recites prayers in a whisper over the altar, facing ad Orientem. Roles are determined by constitutional norms (Church decrees, including Apostolic Constitutions, and custom), not by spontaneous, arbitrary whim. The individual lay assembly member actively participates — in the duly specified way, just as school children (discipuli, “disciples” in Latin) participate in school assemblies, or as private citizens participate (through their representatives) in the state assembly.
So, the notion of Church as assembly is in fact wholly orthodox, and works quite well with the notion of liturgy as “court ritual.” It doesn’t work very well if what you want is spontaneous worship where clergy, laity, or both are free to deviate from procedure at will. If you’re especially keenly devoted to understanding the Church as an assembly, consider the rubrics as proper parliamentary procedure.
* I don’t necessarily mean Modernist in the strict heretical sense. The Church recognizes the category “smacking of heresy.” Something can be literally correct — orthodox, strictly speaking — but simply sounds wrong. I don’t know to what extent the judgment of “smacking of heresy” is context-dependent.
**See the New Liturgical Movement (http://www.newliturgicalmovement.org/2017/01/a-defense-of-liturgy-as-carolingian.html#.WMPxgWciyM9) and Sancrucensis (https://sancrucensis.wordpress.com/2017/01/31/liturgy-as-court-ritual/).