There may not be less than ten [verses for] Malchuyot, ten for Zichronot, ten for Shofarot. Rabbi Yohanan ben Nuri says: If one said three of each, he fulfills his obligation. (Mishna Rosh Hashana 4:6)
Mussaf for Rosh Hashana is unusual. Where else do we have Biblical verses inserted into blessings? Occasionally we see a single verse being inserted as a proof-text, for example in the Aneinu prayer for fast days ("Answer us before we call out, as it says: It shall be that before they call out I will answer them"). But once you have one good proof-text, there is no need for another one. There is certainly no need for a set number of verses - 10 in our case. In fact, according to the Mishna above, it doesn't even matter which verses you choose (within reason) as long as there are 10 of them! Why should there be such an unusual requirement?
The following idea is speculative, but it rings true to me. The different opinions in the Mishna require either 10 or 3 verses. In the context of Rosh Hashana, these numbers do not seem to be random. 10 are the number of shofar blasts we blow after Malchuyot (and Zichronot, and Shofarot). 3 is the number of blasts we would blow, if we knew what a "teruah" was meant to be!
So here is my hypothesis. Once a time, the practice was to recite a verse, and then immediately blow the shofar once. You would recite "With trumpets and the sound of the shofar, blow before the King Hashem" (Tehilim 98:6) and then you would do exactly that. You would blow the shofar along with your Zichronot, making the day a "zichron teruah" (Vayikra 23:24) in the literal sense. And you would recite "The voice of the shofar grew steadily stronger, Moshe would speak and God would answer him aloud" (Shemot 19:19) and then blow the shofar, reenacting the giving of the Torah at Sinai. I think all of Mussaf would gain an extra level of powerfulness if conducted this way.
How does this work out halachically?
Shofar is an unusual mitzvah in that if you blow the blasts one by one, with interruptions between them, you still fulfill the mitzvah. This means that inserting them into mussaf in between verses is not a problem.
A complication arises with the number 10. In theory we blow the shofar 3 or 9, not 10, times for each blessing. This is because we are supposed to do a tekiah-"teruah"-tekiah set for each blessing. We don't know exactly what the required "teruah" is, so we do three different options, one of them tekiah-shevarim-teruah-tekiah (in case the halachic "teruah" is our shevarim-teruah). But this shevarim-teruah is technically considered a single blast, so it would be strange to assign two verses to it, and inserting a verse between the shevarim and the teruah would likely be a forbidden interruption. As a further complication, the number 10 is mentioned in the Mishna, while it seems likely that the uncertainty over the "teruah" did not arise until later.
Let us leave the number 10, then, and move to R' Yohanan ben Nuri's opinion, that only 3 verses are required. These three would match well the tekiah-teruah-tekiah of the basic halacha. An unanswered question here is why some verses would correspond to tekiah, and others to teruah, without a clear justification. A further issue is that R' Yohanan ben Nuri's opinion in the previous mishna is that one recites Malchuyot in the 3rd blessing of Mussaf, but only blows the shofar in the 4th blessing (and 5th and 6th for Zichronot and Shofarot, like we do). So the same R' Yohanan ben Nuri who provides us with the number 3, also disconnects the verses from the shofar blasts!
On the bottom line, I think all these halachic issues can be overcome (with 3 verses being the more likely direction to go in, even though we cannot follow R' Yohanan ben Nuri's opinion across the board).
I thought of this idea before or during Mussaf (I forget which) on the first day of Rosh Hashana this year. It made my Mussaf that day more meaningful, as I envisioned the shofar blasts that could have once accompanied each verse. But it made my second day's Mussaf less meaningful, as I saw Mussaf as a broken version of the original shofar-using prayer, rather than a verbal composition that stands on its own! So I can't really say whether having read this post will be spiritually positive or negative for you. But I think the idea is a fascinating possibility, so here you go.