Prussia was rather marginal in 1813, given how it had been emasculated in 1807.
Napoleon’s defeat resulted from a combination of factors, one enough being not enough :
- first of all, Russia immediately mobilized its full potential in the first half of 1813 instead of taking time to cure the terrible wounds it had just suffered (most of which were self inflicted through scorched earth policy).
In a reality Russia could not "mobilize its full potential" in the beginning of 1813 by a number of the objective reasons:
1st, Russia did not have universal military service that would allow to conscript the whole age group (at least on paper) as was the case in France. The new troops had been levied by conscripting certain (rather low) percentages of the available male population and sharp increase of that percentage would hurt the serf-owners.
2nd, there were limited resources for training the new recruits (IIRC, a reserve battalion per regiment) and additional experienced cadres could be obtained only by taking them from the fighting army.
3rd, situation with the prepared officer cadres was even worse: there were only very few officer schools and, unlike French army, promotion from the ranks was an exception rather than a rule. Of course, there was always a possibility to use enthusiastic young nobles without any military education as the junior officers but their immediate value was going to be quite low.
4th, the new troops had to be armed and by the end of 1812 Russian military production could not provide enough weapons even for the troops that had been raised. There were, IIRC, significant imports from Britain but than, again, could Britain produce hundreds thousands muskets within couple months?
5th, both new and the existing troops had to be dressed: by the end of campaign of 1812 as far as the uniforms were involved Russian troops were only in a marginally better situation than French (mostly because they knew better how to use what's available). All these uniforms and shoes had to be produced.
6th, while it did not lose as many horses as the French army, the losses still had been high and had to be replaced. Immediate great increase of army size meant a need of additional horses in the tens of thousands (cavalry, artillery, baggage train) and you could not just use the peasant horses; they had to fit quite definite qualifications. Strange as it may sound, these horses were not growing on the trees even in Russia (well, at least not in the numbers needed

). They had to be obtained from the breeders and brought to the army. As far as cavalry was involved, they also had to be trained to act in formation.
- secondly, Napoleon and his allies and vassal States lost a considerable share of their war horses in the 1812 Russian campaign. And you can’t replace war horses the way you can replace rifles, cannons or uniforms.
Actually, you can't easily "replace rifles, cannons or uniforms": all these items has to be produced (BTW, "rifles" is a wrong term) and that means that you have to have industry capable of producing them within very short period of time or you have to have huge stockpiles of them. Neither of which is quite trivial task.
(bonus if you can guess which country was the main purveyor of horses. I can give you a clue. It was a very very very big country, in fact the biggest country on earth, which it still is today).
AFAIK, Russia was not exporting horses in the noticeable numbers and, while there was plenty of horses, those considered fit for the military service represented just a tiny percentage of the total. Those who could afford it, had been buying the high quality horses abroad (the British ones had been highly valued). Time when more good quality horses became available domestically was yet to come: modern Don breed became available in 1830's - 50's and those from Caucasus and Central Asia were not available in the needed numbers before these regions had been conquered.
Napoleon's army had been getting its horses from the conquered territories (Germany being one of them and it was producing very good horses fit for the military service) and the main reason for him not pursuing the defeated enemy energetically enough in 1813 was not as much an absence of the horses (if it existed) but a general inexperience and ill-preparedness of his newly raised troops: the boys had been called to the service and they simply did not have enough stamina to perform the same way as the grown-ups..