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In every example of 6/4 time I see:
What decides that this is the way the six beats should be grouped? Is this just a convention?
Why should there be an emphasis on the fourth beat as well as the first, and how would one annotate a time signature where there were six quarter-note beats per bar with an emphasis only on the first beat?
Yes, it's by convention that there are two groups of three beats. The two emphases, however, are not equal: beat one is stronger than beat four, which is what differentiates 6/4 from two measures of 3/4.
Other emphasis patterns can be specified by placing an accent mark over the relevant notes/beats, or a compound time signature can be used. However, the specific case in which there is only one pulse with emphasis would need to be specified by an explicit instruction: "only the downbeat should receive an emphasis; all other beats should be equal", for example.
Music tends to group its rhythms in internal cells of 2 or 3 notes. There are exceptions, but not many. The finale of Stravinsky's 'Firebird' is a '7 on the floor' 7/4 with no internal subdivisions. Brutal, powerful, and pretty well unique! Something similar can happen in pop/rock music. The 'Sweet Caroline' riff comes close - it's 4/4 but just stomps along with an equal accent on each beat. You don't need a special time signature. Just accents on every note! For a more lyrical free-flowing stream of notes the classical term might be 'Senza misura'.
3/2 is three groups of two, 6/4 is two groups of three. Same as 3/4 and 6/8.
3/2 seems the logical way to notate three half-note beats. 6/4 for two dotted half beats is less intuitive. Just let it be the 'not 3/2' way!
I'm slightly surprised this method of writing key signatures (below) hasn't caught on more. I suppose it is rather visually confusing. And the 3/4 6/8 convention isn't hard to absorb. Also, the modern way is often to be pretty fluid with rhythmic groupings, and not to change time signature every time they change. Just indicate them with beaming, accents etc.
It is convention.
If the top number is a multiple of 3 and greater than 3 - 6, 9, 12... - the meter is compound and beats get subdivided by three and the lower number is the note value of the beat subdivisions.
So, 3/4 is simple meter, but 6/4 is compound.
...a time signature where there were six quarter-note beats per bar with an emphasis only on the first beat?
Technically you can't, because you specified the quarter notes are beats. To write that meter it would be 6/4. But, the 6 gets treated as compound. You want a meter that is both simple and has only a single accent on beat one. The only simple meters that do that are those with 3 as the top number: 3/2, 3/4, 3/8, etc.
If you drop the requirement that the quarter note gets the beat, then 3/2 will work. 3 half notes per bar, only the first beat is accented, subdivide the beats for 6 quarter notes per bar.
An overarching concept in metered music is accenting groups of 2 or 3. When division by 3 is possible, you potentially have a compound meter, a meter with groupings of 3, but you need the meter to define that. The classic case is 6 notes. It could be eighth notes in 3/4 or 6/8 depending on whether you want groupings of 2 or 3.
Any time you subdivide by 2 or 3 there is an implicit accenting on the first note of the group. Quarter notes in 3/2 are grouped in 2 and implicitly the second of the two will be weaker. Think of each beat as a down stroke and so the first quarter note of each pair is the strong down stroke and the second quarter note is the weaker up stroke. The beats are strong weak weak. But each of those has a weaker up stroke.
One strong beat followed by 5 weak beats does not follow the convention of grouping by 2 or 3. The system of meters is basically telling you to pick either of those groupings, but not allowing for a grouping of 6. Meters won't express that so you need special instructions of some kind. Unmetered notation probably would not be sure to work, because players will then look for patterns in 2 or 3. Meter 1/4 would be all accents, not exactly right. Probably the simplest thing would just be 6/4 or 3/2 and written instruction to play no accents accept on beat 1 or where an accent mark is given.
To answer your last part - 3/2 works well. Just like the more common 3/4 gives 1-2-3 - 1-2-3, with emphasis only on 1, so 3/2 will have the same effect, albeit with each beat theoretically twice as long - dependent on tempo, of course.
On to the first part - yes, it is convention. there needs to be a way in which those 6 crotchets can be split into two parts per bar, as opposed to three. And so we use 6/4. 1-2-3-4-5-6.
And, the same applies to 3/8 and 6/8, which is more common. Think of 6/8 as two lots of 3/8, but the second bar has a slightly less emphasised first beat than that in the first bar.
In 6/4 time, it's actually the 6 on top that decides how the beats should be grouped, and it defines that there should be a stronger accent on the 1st quarter note and a weaker accent on the 4th quarter note.
The 6 denotes that we're using a compound meter, where each beat is split into 3 sub-beats instead of 2. In reality, 6/4 has only 2 beats, with each beat being 3 quarter notes long. It's like 2/2 time (complete with emphasizing the 1st beat the most strongly, the 2nd beat less, and all sub-beats even less), except each beat consists of 3 quarter notes instead of 2.
Other examples of compound meter include 9/8 (3 beats) and 12/16 (4 beats; think 4/4 except each beat consists of three 16th notes instead of two 8th notes).
As for "how would one annotate a time signature where there were six quarter-note beats per bar with an emphasis only on the first beat?", I'm afraid there may be no such time signature. Even 3/2 time places a stronger emphasis on the 1st quarter note and weaker emphasis on the 3rd and 5th quarter notes. If you want to pull off "six quarter-note beats per bar with an emphasis only on the first beat", I'm afraid you'll need a measure (likely 4/4) consisting entirely of a sextuplet of quarter notes.
I feel like I'm answering this same question every other week. 6/4 is related to 2/2 time. The only difference between 2/2 time and 6/4 time is a dot for the beats. That is the difference between compound and simple time signatures . Simple time has beats without dots and compound time has beats with them.
So 6/4 does not have 6 beats. There is also nothing particular that leads you to subdivide the beats into three parts. 6/4 time is just a time signature with two dotted minims for beats. It is compound duple time with a beat equal to a dotted minim. There are hundreds of different rhythms that can be used. That beat can be divided into just about any number of notes. Some emphasising the beats and some making the division of time unclear or vague.
