Why did the third estate pay most of the taxes even though the higher classes had most of the money?

First, it
should be noted that Early Modern France did not practice a class system as we understand it
today. The modern class system is based in income. The estates were much more
heterogeneous.

The First Estate was the Church, including everything from the
Bishops to the Parish Priests.

The second estate was the nobility, which
likewise contained multiple threads and could include courtiers, the officer corps, the
magistrates, and the courts, as well as impoverished country nobles, who could be just as poor
as the peasants they lived beside.

We likewise see great degrees of
heterogeneity in the third estate, which contained the vast preponderance of the French
population. That being said, the French monarchy certainly left a lot of money on the table, and
this contributed greatly to France's fiscal crisis.

One thing that needs to
be recognized is that governance in early modern France was largely relational in nature, played
out between the monarchy and all the various subsets of the French population.

Furthermore, be aware that if you ever read Absolutist thinkers such as Jean Bodin,
you'll find there is a focus on the importance of custom and tradition. Good rulers were
expected to respect traditions and honor them. Indeed, the early moderns themselves drew this
distinction between what they understood as legitimate Absolutist-style monarchy, which was
contrasted against despotic or tyrannical ones, in which power was unbounded by these
traditions.

Thus, while the Absolutists could be quite autocratic, they also
had to be wary about breaking too sharply with the traditional rights and privileges on which
their own legitimacy and effectiveness as rulers ultimately rested. Indeed, this is precisely
the reason that the estates general was finally convoked as a means to create systemic
reform.