More 'Symbolic Trinitarianism'

Imagine that, in a General Election, each of the three main UK parties gets exactly one third of the vote. As a result it is agreed that the three party leaders (Theresa May, Jeremy Corbyn and Tim Farron) will share the role of Prime Minister. Not only that, but, by some miracle, it turns out that all three of them agree 100% on every single issue imaginable so that every action or decision of one of the three is fully concurred with by the other two. Furthermore, due to the wonders of modern technology they are constantly linked to each other by an online connection which means that any conversation one of them takes part in is instantaneously seen and heard - and, naturally, concurred with - by the other two.
In the situation described above, would it be legitimate to say that although each of the three party leaders is Prime Minister, there is actually only one Prime Minister because each of the three share exactly the same power and knowledge and - because of their absolute agreement on all possible issues - there is no possibility of conflict, rivalry or disagreement between them?
I think that this type of model is one way to make some kind of sense out of the doctrine of the Trinity. Each of the Divine Persons shares the one office of God (ie. the office of supreme ruler of the universe). Since each of the Persons fully possesses all the divine powers and attributes (ie. each is fully omnipotent, omnipresent, omniscient, omnibenevolent etc.) and since, because of the infinite nature of the divine attributes, each individual Person has these attributes to the exact same degree as do the three of them combined, and since there is no possibility of disagreement or rivalry between them, perhaps it is legitimate to symbolise this situation by referring to each of the Persons as God while simultaneously asserting that there is only one God.
A similar model is to think of God as a Composite Being comprised of three inseparable Divine Persons and to assert that each of the Persons is fully God even though none of them is all of God. Again, this is to symbolise the fact that each each of the Persons fully possesses all the divine powers and attributes (ie. each is fully omnipotent, omnipresent, omniscient, omnibenevolent etc.) and that, because of the infinite nature of the divine attributes, each individual Person has these attributes to the exact same degree as do the three of them combined, and also that there is no possibility of disagreement or rivalry between the Persons.
Personally, I incline more towards the first of these models because of my (biblically based) belief that the Father is "the only true God" while the Son and Spirit share fully in His divine power and attributes.
Of course, neither of these models can explain how God could literally be both one Being and three Persons simultaneously. For example, using the first model, imagine one of the Prime Ministerial Persons has an egg thown at him/her. In that case, it would be possible to say that the Prime Minister both did and did not get splatted with egg yolk (depending on which of the Persons one is considering). Yet it is logically impossible for the same thing to be both 'X' and 'not X' simultaneously.

Group names and collective entities

Most of the top premier league football clubs have massive squads of players as well as many non-playing staff. But when 11 members of, say, Manchester United, travel to another team's ground for an away match, we say that Manchester United (not 'some of Manchester United' or 'part of Manchester United') have come to play.
Similarly, imagine a pop group (let's invent one - we'll call them 'The Bloggers') has five members. If one of the members gets sick and the remaining four do a gig, we still say that the Bloggers (not some of the Bloggers or part of the Bloggers) are performing.
If Mr and Mrs Smith and two of their four children turn up at my house I will probably inform my wife that "the Smith family have arrived", rather than denoting them as 'some of the Smith family' or 'part of the Smith family'.
In these cases, a part of the collective entity (team/group/family)possesses all the essence of the whole and (in the case of the football team or pop group) is authorised by the whole to use the official name of the whole.
How far these examples are useful to understanding the Doctrine of the Trinity (by demonstrating that a part of an entity can be legitimately designated by the same term used to denote the entity in its entirety) is debatable, however. I'm not sure we would ever call one footballer by the name of his team or one pop group member by the name of the group. And if Mr Smith popped over for lunch I wouldn't describe my guest as 'the Smith family'. It seems that what these examples consist of is group names, and that such names can be equally applied to the whole of the group or to a part of the group but never to a 'non-group', ie. they cannot be applied to an individual (such as a Person).