I think Germany and Greece can get a pass without this mechanic. Germany because Magna Germania was a thing long before Napoleon and so if not politically "Germany" existed culturally and independently for some period prior to Frankish rule. And although on principle Greece was extremely similar to Italy, the identitarian resonance of Italian city states is just that much stronger to facilitate the model. Ask an average player what type of city state Corinth or Thebes should be and they probably couldn't tell you.
So what distinction are you making between Germany and Italy? Both had common language and culture but no common political unity (after the Roman Empire, anyway), and the individual 'political units' within each had distinctively individual character. The biggest difference, in fact, was that Germany had a smaller population density than Italy, so that many of the 'city states' of consequence in Germany had provincial names rather than city titles: Saxony, Bavaria, Rheinland-Pfalz, Prussia, and only a few were single City States: Frankfurt-am-Main, Hesse-Darmstadt, Mainz and Koln (last two: Electorates). For game purposes, though, they all functioned like City States, unable to 'expand' and control the entire German Cultural/linguistic realm until Prussia managed it in the 19th century. Even then there were still 'internal' distinctions among the 'states' of Germany that persisted: the Bavarian Army was administered separately from the 'German' Army right up to World War One, 50 years after they were supposedly and integral part of Germany.
Not to be too much of a cynic, but I challenge you to find an 'average player' that could tell you what type of city state Milan or Bologna were, or to name more than three Italian Renaissance City States even if you spotted them Venice. As a one-time history teacher, I can tell you it is almost impossible to overestimate the historical illiteracy of the 'average player'. CivFanatics' are definitely not the average player group.
And I think part of this is just a consequence of history and how VI chooses to define facets of civilization; much of the mechanical differentiation between districts is rooted in the Renaissance and Industrial eras, not the classical era. So I think Greece not quite excelling in the city state model might just be an unfortunate side effect of existing too early (same thing with Phoenicia). I think a game which emphasized differences on a classical/ancient scale might have been more facilitative of the idea.
Have to respectfully disagree: most of the 'mechanical differentiation between districts' is utterly artificial and designed purely for game mechanics and with almost no basis in reality. Just for examples, 'Holy Site' locations for Temples or buildings for religious observance were almost always in the center of a city, not separate from it. 'Encampments' as centers for military training were temporary sites, except possibly for the 'Field of Mars' next to Rome, unless they were permanent billets for armies like the Roman Castra or Border Forts. Heck, the Barracks as such wasn't used in Europe until the very late Renaissance, and even then, for instance in Prussia, they were scattered throughout the towns and cities as individual rooms for troops, not centralized areas.
This is not to criticize the District concept: once the decision was made to spread out installations on the map, they had to have some 'structure' for placing Buildings and Bonuses, and given the huge range of possibilities in historical cities from a huge variety of cultures and Civs, there was/is bound to be a lot of artificiality to whatever they came up with. Thinking that the Districts are specific to anything historical is a mistake, though: they are artificial constructs that we can, therefore, adapt to whatever Game Mechanic we want to implement.
As for your observations on unification and the piazza, while I don't want to presume Vatican dominance or throw out the piazza idea, I had a conversation elsewhere about "liberating" cities into city states that developed some interesting design space. Much like how Venice took over city states with trade, there are several ways a central city state could "convert" city states into joining their civ.
I agree that there could/should be a variety of ways to 'include' City States into a a Civ, but in the specific case of turning a Civ composed of culturally related City States into a 'normal' Civ I think there has to be a method of indicating which City States are related enough to include. Venice is a good example: she dominated and controlled a number of cities outside Italy (some Far Outside), but none of them outside Italy proper ever became part of an Italian State. In fact, just about the only indicator left of any Venetian influence in most of them is the occasional 'Venetian lighthouse' as in Khanea in Crete. Likewise, the many 'Greek City States' founded outside Greece all became parts of other States: Italy (Syracuse, Taranto), France (Marsalla), Turkey (the 'Ionian' cities).
We could simply use the current 'Loyalty' mechanic, which is directly related to proximity, and include a Cultural Component (including, as you state below, Religion) which would rather neatly relate the conversion of City States into a Unified Civ to also having a contiguous 'cultural/linguistic/geographical' set of borders.
To my mind, if it were Vatican City State leading this proposed civ, then it could acquire city states by simply converting them to its religion. We already have the missionary mechanic in place so most of the development work is done. And an alternative means of establishing control in city states could be to have a UB in the Basilica as a means of spying on or influencing cities outside of your empire.
IF the Vatican City State was to use a religious conversion mechanic, then either the City State would have to have a guaranteed religion, like Arabia now perhaps, or a serious, almost automatic Religious Bonus from the start, or it would wind up being a Dead End Civ whenever it 'misses out' on a Religion.
I am not sure how I feel about how the piazza should be implemented. On the one hand flavorfully a piazza doubling up on district typing would create specialized powerhouses. On the other hand, I wouldn't want to make city states so lopsided as to be unsustainable. The piazza would likely need some general bonuses as a balancing measure. So perhaps it provides additional culture/production/whatever adjacent to a specific district, but also universally provides some housing and food?
Agree. IF we differentiate City States from Civ Cities by allowing them only one District, then certainly that one District has to be multi-purpose or the City State winds up , as you say, 'unsustainable' or wildly unbalanced.
On the other hand, we could get some of the same effect by simply making it harder for City States (of any kind) to build Districts: perhaps requiring a 5 Pop per District instead of 3, which for most City States would allow them to build only 1 District until the Classical Era or even later, depending on their position and terrain. Then make the 'special' District for that City State available early or first, and that District sets the 'tone' for the City State and also indicates which 'Culture Group' they can be unified with.
This is simply Mechanics within the Civ VI framework. Right now City States are distinct from Civs and don't join them except by Conquest. Once we decide that they may be 'specially founded' by a given Civ and eventually 'voluntarily' join a Civ or Form a Civ, and therefore should have Special Characteristics somewhere between those of a 'normal' City State and a Civ City, like being able to build Districts at all, the rest is hashing out the details and mechanics of how to make it work in the game . . .