What if Cortez lost to the Aztecs

In OTL the Spanish renegade Hernando Cortez conquered the Aztecs after a short, but bitter war. The Spaniards were incredibly lucky as smallpox ravaged the surrounding lands. But what if the Aztecs defeated the Spanish and maintained control of Mexico.
First there must be a suitable turning point inwhich history can change. The most suitable turning point could the Battle of Otumba where Cortez, a handful of Spaniards and few thousand native allies fought of a 20,000 strong Aztec Army which proved to be the turning point of the war.
Let us say Cortez loses and is killed by vengeful Aztec warriors. This would end all Spanish attempts to conquer the Aztecs. Smallpox would still ravage the empire, but it would return to normal in a few years as the Spanish would not be able to take advantage of it. Smallpox would also ravage the Maya and Inca, but they too would recover. Athualpa would become the Sapa Inca and would probably crush revolts and expand Inca territories further.
The Aztecs under Tlatoani Cuitlahuac would forever nurse a grudge against the Spaniards, but they had no naval capabilities and so, would have to contend with fighting Spanish raids. These Spanish raids would have been born of the desire to avenge their comrades.
The Spanish in Cuba would conquer the surrounding islands and spend their time exploring the surrounding lands, but a new power would come to play in the region. France
France did not have the colonial head start of the Spanish, but true to form, they would attempt to find allies to combat Spanish influence in the region. In Europe the Italian Wars were raging and the only thing keeping Spain fighting was the gold in our timeline. In 1526, the War of the League of Cognac flared up and without the Gold, Spain and the Holy Roman Empire would not be able to contend with France, England and a coalition of city states.
For the sake of it, let us say France wins the war and gains Savoy, Lombardy and Naples while England gains money and the Italian city states would gain freedom from German control. Spanish power in Italy is broken and their revanchist leadership begin to plan revenge.
They would look westwards, and colonise Florida.In an attempt to curtail Spanish ambitions, the French would ally with the powers in the Caribbean, such as the Aztecs, various American tribes, and Maya kingdoms. The French would institute an trading system where in exchange of muskets and cannons, the natives would trade their profitable spices, plants and gold with them, with the French now holding a total monopoly over spices.
Charles V would colonise places like Venezuela and now German, not Spanish colonisers would represent Hapsburg interests in the region. Meanwhile France would also instigate the Taino Caciques into revolts and give them weapons as well.
Finally, this whole thing would boil over into a new so called Italian War, just this time fought on a more global scale. Spain would attempt to retake Italy with an invasion from the south while the HRE would attempt to do it with one from the north. Many battles would be fought with France being invaded by Henry VIII as well. Battles would erupt in the Low countries, Rhineland, Italy and the Pyrenees.
With bloody battles intensifying and France finally about to get overwhelmed, they would play their trump card. The European- like Aztec and Maya Armies would cross the seas on ships, provided or built with the help of France. They would take Cuba in a surprise attack and he take Hispaniolawith the help of Taino rebellions. At this point, the Caribbean and Central America would be redrawn.
The Tainos would rule both Cuba and Hispaniola under a united empire and the Aztecs and Maya would get the spoils of war, along with the French. Klein-Venedig and Florida would be taken over by France and turned into a independent kingdom. While at first glance, this may not seem to br a huge difference I would set of a chain of events.
A classic sunset invasion in support of France seems out of the ream of possibility but with the loss of revenue, the war efforts of Spain and the HRE would crumble, with new revenue flowing into French coffers, an attack would knock Spainout of the war with the HRE crumbling soon after. The French would gain control of the Three Bishoprics, Flanders, Sicily and Sardinia.
England would sue for peace, giving a hefty amount to France as payment. As a result, Spain would lose it’s head start while France would gain control of the Americas in the future decades.
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