The Archbishop of Mainz had extraordinary secular powers from the creation of the Holy Roman Empire, until the Diocese's decline in the 15th century. Mainz was the most important diocese, being called primas inter pares, first among equals of the German bishops. It also frequently held the title Primas Germanie, most important German bishop. As a prince, it collected taxes in its territory.
It had a crucial role in the formation of the Holy Roman Empire. In order to understand its importance, it helps to go back to the end of East Francia and the creation of the Kingdom of Germany. The Kingdom of Germany became the core of the Holy Roman Empire, which included the Kingdoms of Italy, Burgundy, and Bohemia.
Liutbert was archbishop of Mainz from 863-889. He was probably the first to be called Archchancellor. He oversaw the continued works on the Annals Fuldenses. He put down a rebellion of Moravians and Serbs in 871. He organized defenses against Viking raids into East Francia.
Hatto I was bishop of Mainz from 891-913. He became acquainted with the king of East Frankia, Arnulf, who appointed him as Archbishop of Mainz. He became so close to the king as an adviser that he was called "the heart of the king". When Arnulf died in 899, Hatto became regent to his son, Louis the Blind. He remained influential throughout Louis' reign.
Hatto became involved in the dispute between the Conradines and the Bambergs in the Duchy of Franconia. Henry Babenberg had been granted the Duchy of Franconia by Charles III (the Fat). Henry had been the Commander in Chief under Louis III and Charles III. Hatto supported the Conradines against the Babenbergs. In a battle in 906, Conrad the Elder was killed, as was two of the three Babenberg sons. Louis III then intervened; notably, his mother was a Conradine, and the last Babenberg was captured. Hatto ignored a promise of safe conduct, captured and executed the last Babenberg. Conrad became duke of Franconia. Hatto was prominent in securing Conrad as King after the death of Louis the Child. Hatto supported Conrad against the Saxon, Henry the Fowler. The Saxons saw Conrad as an extension of Hatto's interests. Conrad was the first non Carolingian King, and the first to not be elected by the nobles. This marks the end of East Francia and the beginning of the Kingdom of Germany.
The archbishop Frederick plotted to asassinate Otto I. It's not suprising, then, that the next one, Wiliam, was Otto's son. Willigis was archbishop from 975-1011. After the death of the Byzantine queen Theophanu, he adopted Otto III and served as regent, along with Adelaide of Italy.
Aribo, (1021-31) developed Thuringia by minting coins at Erfurt.
Siegfried II (1200-1230) obtained the right to crown the King of Bohemia. This was retained by the archbishops of Mainz until 1343. This later date was the highpoint of the Diocese. A war over the position of Archbishop from 1461-63 devastated the city and Diocese, which lost a lot of funds and territory. This marked the decline of the Diocese.
These are highlights of when it exercised great influence. More often than not, these early archbishops were an antagonist to the Crown. They frequently backed other stem (tribal) duchies in rebellion. You have listed the formalities that the Archchancellor retained in the later period, but it was past its prime. Its continued importance would be as a Church representative, not as a powerful prince of its own territory.
To clarify, the Archbishop of Mainz was only the Archancellor of Germany. The Archbishop of Cologne was the Archancellor of Italy.