usually the argument is put as if the danger is "theocracy", but there is a difference between the church power structure running the government (as in Islam) or allowing the common rules of the people, including those of people who are church members, to have input into civil law.
For Catholics, this goes back to the middle ages, and was one reason for the schism when there were two popes (one controlled by the French king) and behind the martyrdom of Thomas Beckett...
Nowadays, we see the problem in the UK, (or Scandanavia, or Catholic France and Germany) where the "official" church has bishops trying to go along with the government/ popular opinion of non church goers, to change their dogma to conform with the PC who don't attend church, while their non national members say: WTF ARE YOU DOING?
Even in eastern Christianity, the emperors often told the church what to do in their dogma (and bishops like St Athanasius opposed them, facing not just unpopularity but exile). A lesser version is to do like Stalin, and make sure all the bishops are KGB members and get rid of the priests who might be a threat to the atheism of the regime.
So it bodes to remember also Marie Antoinette, who was slandered by those trying to change the regime, and how she faced death bravely, without benefit of clergy, because she refused to have a true priest risk his life by giving her the sacraments, and refused to see a priest who was untrue to the faith...
Heh a least Anne Boleyn, was allowed to be "comforted", albeit by the turncoat bishop who refused to defend her when he knew she was innocent, because to take a moral stand against the king's wishes might jeopardize his larger agenda, "reforming the church".
Yet in the French revolution, it was not just a gradual takeover of churches beloved of the people, with minor changes of personnel and dogma...at the time of the French revolution, the church was taken over by unbelievers and forced to become a puppet of the government.
TeaAtTrianon has the story: early "Reforms" which let the (bankrupt) government confiscate the church properties and ask church to obey government decrees were reluctantly approved by the king, but of course, things didn't stop there: they then decided the priests had to make an oath to obey the government, and almost half did not. The result was schism.
Later on, persecution followed, but it all started with a reasonable request that the church "cooperate" and obey the government.
Crisis magazine report here on history, and how what Catholics are facing now could be a similar takeover.
Senator Kaine’s attempt to enshrine the laws of the United States above those of his own faith and conscience recall the juror priests of France who betrayed their religious vows for the sake of self-preservation as agents of the state. In this regard, we must ask ourselves to what extent Senator Kaine would be willing to weigh the laws of the state over his personal beliefs and religious convictions. Would he allow that there is a limitation to the law’s reaches, and if so, who or what is its arbiter?
Note that Crisis magazine is "more catholic than the Pope", but nowadays that doesn't say much...
This is about the danger if the Pope allows China's subservient pseudochurch to become the one recognized by the Vatican, effectively throwing the underground church "under the bus" in the name of "unity" or something...at a time when that government has increase the persecution of believers.
Christianity Today article on this.
Christian Post article points out it is not just Christians:
Last year the Chinese President Xi Jinping gave a major speech on religion in which he argued the "management of religion is in essence the management of the masses"
China Aid warned that it is not only Christians who are being persecuted, noting that followers of Tibetan Buddhism and Islam "likely experienced even more intense government persecution" at the hands of the atheistic Communist Partyand no, this is not about Hillary's minions infiltrating the Catholic SJW movement.
nor is it about Switzerland and Canada threatening religiously run old age homes that refuse to euthanize their patients, or the umpteenth ACLU lawsuit trying to force Catholic hospitals in the US to allow sterilization.
This is not just about the "reformers" trying to co-opt a powerful church to assist them in their goals. This is about a long war in deciding who do you serve, the government or God?