I was shown the Chasam Sofer (Gloss to Orach Chaim 59:3) brings the Tur @DoubleAA mentioned as the source for the custom, and adds that we apply it at this point in the prayers because if we were to slur our words, it would sound like וממליך מת, corronating the dead. Since there are indeed those who worship the dead, we ought to be extra careful to avoid this phrase.
The Siddur Tzelusah D'Avraham has a commentary on the Siddur called "Emek Bracha". This Siddur is based on the practices of Rav Avraham Landau (1784 - 1875) and it was put together by his grandson Rav Menachem Mendel Chaim Landau. It writes as follows:
וממליכין את שם העולם נהגו להפסיק בין וממליכין לאת שם ולהמתין על הש"ץ והוא על פי הטור שכתב בסימן ס"א שלא לתכוף וממליכין את שם עיי"ש והטעם כדי שיהיו כל הקהל עונין כאחת את שם האל הגדול כו' והא דרך שירה וכמו הקדושה עצמה
We crown the name. People are accustomed to pause between "וממליכין" and "את שם", and to wait until the prayer leader. This is based on the Tur (OC 61) not to be too quick to say וממליכין and את שם. And [another] reason is so the whole congregation says together "The Great Name etc." This is in the manner of a song of praise, like Kedushah.
So, besides attributing the reason to the Tur, as @DoubleAA suggested, he adds that there's an element of "song" to this custom.
Like DoubleAA, I noticed many older siddurim don't have a paragraph break. Many had a period or colon before את שם, probably in deference to people pausing. We need more evidence how old this custom is, to compare to how siddurim print this line. From the above source, it's at least approximately 100-150 years old.
What's ironic is this siddur is nusach Sefard, so they say וממליכין, making it a little less problematic to say the word את immediately after. Maybe that's why he suggests another reasoning.