It is indeed true that the tsar wanted Charles to be the chief military command of the Coalition armies. Why this was and why it didn't come to pass are pretty complicated, though.

It's true that Austria was the last great power to join the Coalition; while you present it as a detriment, really, it was a benefit; after the campaigns of 1812 and Spring 1813, the Russian and Prussian armies were exhausted, having crossed half of Europe and lost two major battles against Napoleon in 1813 alone. Austria, meanwhile, had undertaken relatively minimal exertions in this time, limited to the Auxiliary Corps's campaign in Russia; while it did undertake some severe combats in this time, there was nothing on the scale of Lutzen and Bautzen. By the end of the summer armistice, the Allies and the Imperials were relatively evenly matched, the advantage laying with Napoleon if anyone. Both sides thus sought to court Austria, hoping to tip the scales in their favor. Robert Thomas Wilson was a British attache who served in a couple different headquarters during the Wars of Liberation, and had the occasion to speak with the king of Prussia during the Allied advance in 1813; he expressed concern that Prussia lacked the steadiness and Russia the power to meet the occasion.

“It is Austria – Austria! – which holds the balance, and it is to Austria that we should direct all our thoughts, whilst we afford sufficient succor to Prussia to keep her vessel afloat.”

Austria had considerable manpower resources at her disposal; while the army had been capped at 150,000 men by the Peace of Schonbrunn, and the financial situation was precarious, the army had retained the structures and mechanisms necessary for major expansion, and eventually put more than half a million men in the field, all told. During the 1812 campaign, they had fielded an Auxiliary Corps for the invasion, which Napoleon noted contained far more cadres than was strictly necessary for the requested force, as well as a corps of observation in Bohemia. Digging deeper, the empire had retained many companies at half strength, as well as third battalions on cadre status. These could be fleshed out with some speed, as over the 18th century the Habsburg monarchy had developed the key features of an absolutist state, carrying out detailed censuses and inspections of the country. All the residential buildings were numbered; the men tallied up, their trades noted, even the supply of draft animals in different localities was kept track of. Entering the war at a point where its strength could tip the balance, Austria enjoyed a privileged diplomatic position. As Metternich remarked in a letter to Schwarzenberg,

The power that places 300,000 men in the field is the first power, the others are auxiliaries.

So then, why Charles? Well, who else? Charles was Austria's great hero of the French Wars; he won his first laurels fighting in the vanguard at Neerwinden, a smashing Austrian victory against Revolutionary France; he defeated Jourdan and Moreau separately during their 1796 invasion, and drove back Jourdan again during the Austrian 1799 offensive in Switzerland. In 1809 he became the first man to defeat Napoleon in a major battle, driving him back over the Danube at Aspern; during the fighting, he showed the greatest courage, taking up the standard of the Zach Grenadier Regiment and personally leading his grenadier corps in a counterattack against Lannes's breakout. He'd certainly suffered his share of defeats, but these were often the result of disaster befalling his predecessor in command, such as in the campaigns of 1797 and 1800, when he did all that could be expected in such dire circumstances. He was defeated in 1805 and 1809, but he managed to keep his army in being, which is more than most of Napoleon's victims could claim. Moreover, between wars, he had proven himself an able administrator, reforming the bureaucracy of the Austrian war machine, reforming tactics, and reorganizing the army to fight modern warfare. Charles was peerless in the esteem of the army, and the people held him up in plays and poems as the successor to Arminius of the Cherusci, the symbol of German defiance to Imperial conquest. As much as the Habsburg monarchs allowed anyone but them to control the army, it was Charles's army.

All that is to say that it was not any lack of military acumen that kept Charles from taking up command in 1813. Rather, it was Charles' unwillingness to politically subordinate himself to the kaiser and Metternich that burned that bridge. The Main Army would be composed of contingents from every great power on the continent, and would be host to three monarchs. The relationship between kaiser Franz and his brother was strained at the best of times, but giving him a free hand politically in such a delicate diplomatic and military situation was simply unacceptable.

It is the political dimension of war that most recommended Schwarzenberg over Charles. Schwarzenberg was very much a soldier-diplomat, having served in different positions in both France and Russia by 1809. He had negotiated the marriage of Marie-Louise and Napoleon; he navigated the complexities of commanding the Auxiliary Corps in 1812, with Napoleon himself recommending him be made a field marshal, and managed to extract his corps when the campaign decisively turned against France; in 1813, met with Napoleon to attempt negotiating a peaceful settlement, though this was a failure. Even the most mediocre Feldherrs spring from a career of great personal bravery, and Schwarzenberg was no exception, having earned a fine reputation on the battlefield as a leader of men. He was a much more personable figure than the austere, aloof Charles as well; he got along with people, no small achievement when,

“It really is inhuman what I must tolerate and bear, surrounded as I am by fools, eccentric projectors, intriguers, asses, babblers, and niggling critics. Vermin in countless number gnaw at me and torment me to the very marrow of my bones.”

He and his chief of staff Radetzky didn't have much in the way of personal affection or friendship, but even then they kept up a productive professional relationship; there wasn't really any rancor coming from that partnership, and in war, keeping that kind of friction to a minimum is very helpful. Through the campaign, he recognized political questions were the affair of the civil government as well as the army, and consistently deferred to Metternich, who had argued for Schwarzenberg's appointment on the grounds that 'he makes war, but is not a politician.'

This doesn't really factor into why Schwarzenberg got the job, but it is interesting to note that throughout the 1813 campaign, he arguably demonstrated greater strategic ability than Charles had during his campaigns, and indeed most of Napoleon's enemies. Charles and most Allied generals had grown up in a school of warfare that very much emphasized positional strategy, and conceptualized victory and defeat in terms of the possession of discrete geographic points, such as fortresses, magazines, lines of communication, and terrain features. This would coalesce into the 'key to the country' concept in Charles' postwar writings, or the idea that nations had geographic points which, if held or taken, ensure the preservation or conquest of its territory. Clausewitz is pretty dismissive of this concept on its face, but it must be said that the past quarter century of war powerfully demonstrated the preeminence of armies and battle over geography as war's center of gravity when grave interests are at stake.

The Austrian authored war plan for 1813 is typically framed as an 18th century style positional strategy of exhaustion, but I've found this to really not be the case. Reading Schwarzenberg's letters, I was struck by how little he seemed to care about geographic points; rather, the enemy army and preparation for decisive battle are his chief fixations. He measures success and victory in terms of damage inflicted on the enemy, in battle casualties, in prisoners, in captured guns. The means of damaging the Imperials was often privation, rather than combat, but this is an accidental, rather than essential distinction from the classic strategy of annihilation. Throughout the campaign, these losses were considered preparation for the decisive battle, which the Allies sought to fight when they achieved 2-1 superiority. Schwarzenberg submitted his plan to link up with the Army of the North to the Allied council of war on 8th September, in which his and Bernadotte's armies would join on the fields of Saxony around Leipzig and force Napoleon into a battle on reversed front against great superiority of numbers. The plan was modified when Blucher refused the passive role he was assigned covering the Bohemian mountains, but the key decision of the campaign demonstrated an essentially modern understanding of war, resulting in Napoleon's greatest defeat.