As a brief aside, you should be aware that Tuchman's A Distant Mirror is a particularly flawed work, dated in its own time not to mention ours. She relies heavily on scholarship which is either anti-Catholic or anti-Christian (e.g. Gibbon), and is by her own admission interested in drawing comparisons to her own time, a fact which colors her work. Charles Wood's review of the book in Speculum is long but worth reading for elaboration on these points.

However, the reality your question describes is neither entirely fictional nor wholly inaccurate, and so I will try to provide you a few considerations by way of an answer.

The integration of clergy into the state as functionaries in the West has its beginnings with the legalization of Christianity by Constantine in the 4th century and solidifies with Christianity becoming the state religion of the Roman Empire with Theodosius I in 380 AD. These bureaucratic positions become more important as the Roman state itself goes into decline while the institution of the church survives. Bishops and abbots maintain important roles in the local politics that emerge post-empire, often to one another's frustration. In many ways, the church provides a political network which would otherwise have not existed, as well as a connection to the memory of Rome, which is still fresh and gilded in the minds of many. For a good example of what the role of bishops and abbots looked like in this period, I would recommend taking a look at The Life of St. Colomban by John of Bobio. The account is written in the 7th century about the 6th, so on both accounts it gives a good snap-shot of this period. You can see the ways in which the spiritual and political overlapped as a matter of day-to-day life. It can be found online here.

Not only was the institution of the church an important aspect to politics, it retained an important bureaucratic function as the record keeper. This makes sense, as in many areas knowledge of reading was retained exclusively by the church. Thus over time, records of baptisms and funerals served the function of modern birth and death certificates, respectively. This is one area where the issue of fee charging becomes sticky. We do not think twice that the state taxes us and spends that money on the making and keeping of records, yet there is the problem of simony, that is, the buying or selling of ecclesiastical offices or benefices. The term is derived from Simon Magus, who in the Book of Acts attempts to pay Peter for the gift of the Holy Spirit and is rebuffed strongly. Simony has always been outlawed by tradition and by canon law (which did not do much for it becoming a widespread practice). How does money find its way into church practices, then?

One possible answer is to point back above to the political role of the church. As society becomes increasingly feudal, bishops and clergy maintain themselves through the ownership of land, often on behalf of a higher lord. Such positions are lucrative and carry enormous responsibility politically, economically, and militarily. We see this play out through the Investiture Controversy, which was a dispute over who endows a bishop with his authority, the secular ruler or the pope. Secular rulers wanted control over who was a bishop, as they wanted to rely on their loyalty; the church didn't want its clergy so beholden to secular authority. Point being, these roles were lucrative, enough that people really cared who held them. Acquiring them, particularly from a secular ruler, often required some form of remuneration, as you were basically being given a free income. It became the practice to buy the rights to a particular episcopal seat or parish and then either never do your job (in the case of the bishop) or hire someone to do the actual priestly work for you, thus netting you a profit for nothing. Again, none of this was strictly legal under canon law, and there were many attempts on the part of the church to curb or end this practice (one of the main reasons they opposed lay investiture of bishops), but this a major way in which money begins to change hands in more and more blurry ways.

Another way in which money enters the equation comes with the concept of weregild or blood money. This concept came to Christianity from Germanic customs in which money was often paid as restitution for wrong done. The name points to the fact that it originated with murder, namely that if you killed someone, punishment could be lessened or dispensed with by paying a certain fee to the individual's family. We see this in our own modern judiciary systems where one is punished by fee paying or, if unable, jail time. This concept of paying money to lessen fees found its way into penitential guidebooks (books meant to help priests/confessors assess appropriate penance for specific sins), which as you can imagine, helped to create the notion that relief of penance was somehow purchasable. This will grow over time into the practice of selling indulgences (i.e. remission of punishment for sins), a major money-maker for the church hierarchy.

The two above mentioned sources combined into a potent cocktail where you have a person borrowing huge sums of money so as to buy an episcopal seat, then going on an indulgence-selling spree in order to raise the money to pay back the debt. One example, Albert of Brandenburg, did just this to become the Archbishop of Mainz. In order to foster indulgence buying, regular church services were cancelled or their sermons supplanted by indulgence preachers, and the meaning of indulgences were blurred so as to suggest that they not only forgave time in purgatory but gained salvation as well. This infuriated many, including one junior university professor, an Augustinian friar named Martin Luther who authored a set of ninety-five theses challenging indulgences. He was by no means the first to object to these practices, but the response his complaints received among both clergy and laity should give us some indication that anger over these practices was rampant.

I've gone past your question. Hopefully, this helps you see how a political reality thrust church figures into roles which more and more dealt with the use of money. Though my answer flirts with passing judgment, I think it is far better to try and understand the reasons why this happened and what problems rose from it. The political involvement of church figures accomplished good as well, though it is hard to see or remember in light of some of the abuses, which were many. Institutions are products of their time, even religious ones that we might want to rise above for personal or ideological reasons. In the end, they are made of people, and understanding those people and their situation is the best we can strive to accomplish.

Sources / Further Reading:

The Rise of Western Christianity by Peter Brown (Blackwell, 1996).

Medieval Christianity: A New History by Kevin Madigan (Yale, 2015).

History of the World Christian Movement, Vol. I by Dale Irvin and Scott Sunquist (Orbis, 2001).

The Reformation: A Brief History by Kenneth Appold (Wiley-Blackwell, 2011).

The European Reformations, 2nd Edition by Carder Lindberg (Wiley-Blackwell, 2010).