First let me preface this answer by stating that whilst I am significantly versed in the Wars of the French Revolution, much of what I know comes from a background in the Early Modern period of the Habsburg Monarchy. Second, to fully provide you with a comprehensive answer I need to understand your form. What do you mean by Ancien regime? Below, I take it you mean it to be the nobility or Second Estate and not the actual Monarchy. This is an important differentiation when examining the governance of an early modern power. With this in mind, the ‘miracle’ I describe has more to do with the failings of the First Coalition, rather than the achievements of the new Governments of the French people. However, I’ll attempt to answer the second part of your question without straying too much over the Rhine.

The Bourbon Monarchy had put in place significant reforms which supported the fiscal-military policies of France during much of the eighteenth century. Whilst these reforms did not significantly centralise the collection of tax and men from the French armies, as was attempt by the enlightened absolutist of the Habsburgs, they provided enough to support the army until the Monarchy was forced to call the national assembly. The reconvening of the national assembly was an attempt to levy income from two of the richest estates- the church and the nobility -who thanks to the outdated legislation were exempt from taxation by the Monarchy. Of course, the Monarchy was unable to force legislation which allowed them to tax the First and Second estates, leading to events too numerous to discuss.

For the majority of the eighteenth century the Monarchy was hobbled by the nobility who owed much of the agricultural land of France, which the significant proportion of taxable income was derived from. This land was either farmed by freehold tenants who paid tithes to their ennobled landlord, or by serfs who owed direct services to their landlord. Most, if not all profits derived from such farming went directly to the estate holder, who were mostly if not always the nobility. With the majority of the French economy and the services of its people controlled by a small proportion who made no fiscal contribution to the running of the state, the Monarchy had to rely on other means to support its dynastic goals.

Most of the military income for the Bourbon Monarchy was from tariffs on trade and commerce internally and externally. This was the case with much of Europe, leading to the empire building and mercantilism of the previous two hundred years. When the new constitution in 1789 abolished serfdom and the rights of the nobility, it freed up significant fiscal-military resources for the new French government. Furthermore, the acquisition of much of the catholic churches holdings and wealth also provided the new government untapped resources. Thus, with the removal of the First and Second Estates the new French governments could call upon significantly more wealth then the Monarchy had done in the preceding fifty years.

Much of the social and economic reform you allude to had actually been attempted by the German Princes of the Holy Roman Empire and by the Habsburg Monarchy. Who, either informed by Aufkerlung, the Enlightenment, or the administrative ideas of Cameralism, had precipitated a top down restructuring of their states and societies in support of foreign policy. Such bureaucratic reforms, were bound to flow over borders. Indeed, much of the social reforms of the revolutionaries were greeted with approval by German intelligentsia, who viewed the archaic power of the nobility as unenlightened. Joseph II, was all for a constitutional Monarchy in France, though definitely not in his lands. Indeed, Joseph was a ruler who was very much for reform for the people, just not by the people. And believed so long as the French King went along with the constitution, Europe would more or less remain the same.

To summarise, the Ancien Regime contributed little to the fiscal-military system needed to sustain military efforts and its removal provided the impetus to change much of the bureaucracy of France in defence of the Revolution. However, the defence of the Revolution was not down to the administrative efforts of the clerks, lawyers and merchants which made up the new governments. In fact, it was both the converging and diverging strategic goals of the First Coalition, a problem which has always obstructed successful coalitions. The internal pressures faced by the Habsburg Monarchy, brought about by radical reforms of Joseph II and his alliance with Russia, as well as the primacy Prussia placed on the second partition of Poland in 1795, which prevented a Dual German victory. These points, combined with the fiscal-military cost of engagements and the unwillingness of Coalition Generals to improve operational doctrine, are what assisted the French performance and contributed to the continuing wars of the French Revolution.