Then it is very difficult for me to understand why the pope was not able to or would not consider the possibility to instigate a popular uprising against the King himself.
A popular revolt was not an option in those very pre-democratic times... however, it would be legitimate to ask, whether pope tried stirring population in support of a rival claimant to the French throne.
Assuming this is correct, there's this thing that I want to understand: It looks as though at the time these events took place, the Pope had far better administrative machine and more spiritual authority than the King.
This is not quite true: the trials took place between 1307 and 1314, that is during the papacy split, when the French King installed his own pope in Avignon:
During this time period the power of the papacy had declined and most of the popes of the twelfth and thirteenth centuries found themselves either fleeing Rome or not allowed to enter at all. Also at this time antipopes backed by the German Emperors were common fixtures in the Emperors' bitter struggle with the Church. One of the last thirteenth century popes was Peter Morrone, an old man selected to be pope as a compromise, who as Pope Celestine V proved too old and too ineffective to rule the Church and upon realizing this himself, he abdicated. This caused a tremendous protest throughout the western Church and had a divisive effect on the next pope, Boniface VIII. Pope Boniface was in many ways the opposite of his predecessor in that he was very capable, determined and even bold, but many held that a pope could not abdicate and that Celestine remained the true pope. Boniface in turn captured the old pope, who had sought nothing more than to retire in peace, imprisoning him until his death in 1296. Boniface VIII continued to impose his control on secular authorities, Edward I of England and Philip IV of France, who both protested against his authority, but Philip IV of France proved his most formidable opponent. Philip attempted to tax the church, which Boniface refused, beginning a long series of struggles between the two. Finally in 1303 Guillaume de Nogaret, Philip IV's lawyer, drew up a list of 29 charges including black magic, sodomy, heresy and blasphemy against Pope Boniface. In turn Boniface announced that he intended to place the kingdom of France under interdict. This threat to Philip might have led to revolution so de Nogaret and Sciarra Colonna, leading a force of 1600 men, attacked Anagni where the pope was in residence. They captured Boniface and held him prisoner for three days. After four days, however, the residents of Anagni rose up and expelled the invaders and took Boniface to Rome in triumph. But the ordeal had been too much for the 86‑year‑old pope and he died days later. Philip IV was determined not to have a pope interfere with his plans again and after a year the conclave was still unable to decide, so an outsider was suggested in the person of Bernard de Goth, Archbishop of Bordeaux. He had been a supporter of Boniface, but Philip arranged a meeting promising to support him as pope if he would agree to certain conditions, including reconciliations between France and the Church and absolution for any of Philip's men who had fought and captured Boniface. Bernard de Goth became Pope Clement V on November 14, 1305.
As we see, Pope Boniface indeed tried to stir a revolt against Philip IV but was outplayed and replaced by a more obedient pope.