Did it, after all, arrive in Spain and deliver its treasure to King Charles?

Yes, but...

It wasn't exactly a treasure ship. Not like the treasure ships that would come later. It was more of a down-payment-on-a-bribe ship.

The story the OP and their video tells got a little smashed together and mixed up sending a ship back to Spain with scuttling his ships. Instead I'll draw on The Memoirs of the Conquistador Bernal Diaz, an eyewitness account by Bernal Díaz del Castillo, one of Cortés's men. It goes something like this.

Diego Velázquez de Cuéllar is made Governor of Cuba under Diego Columbus. Veláquez and the settlers don't like being under Diego Columbus, so they form a Cabilo which directly answers to the King of Spain and put Velázquez in charge. Velázquez is now free to do basically whatever he wants.

Velázquez starts chartering expeditions to Mexico. In 1518 it's Cortés's turn and he rapidly builds a large expedition. Velázquez gets jealous and fearful that Cortés will take what he finds for himself, Cortés was also dating his sister-in-law which didn't help, and revokes Cortés charter. Cortés sails anyway, committing mutiny punishable by death.

Cortés stomps around on the Yucatán peninsula a bit shaking down the natives for whatever valuables he could, but not finding enough to excuse his crime. Hearing about the Aztec Empire, and thinking its conquest is his ticket out of a death sentence, he founds/conquerors Veracruz in May 1519 as a base of supply for his coming conquest, and also for a ticket out of his legal jam.

Taking a page from Velázquez, Cortés has a Cabilo declares himself governor of Veracruz. Now he's, theoretically, acting under the authority of the King. But the King hasn't authorized this. So Cortés sends a ship back to Cuba with whatever valuables he has and some trusted men to ask/bribe the King to make his governorship legit. Since this was before Cortés's conquest of the Aztec Empire, there was little treasure and everyone had to empty their pockets for the gift to the King.

The building of the fortress having solely occupied us for a length of time, and now in such a forward state that we could lay the woodwork, we began to grow tired of doing nothing. Almost the whole of us, therefore, addressed Cortes in a body: representing to him, that we had now been three months in this country, and high time we should just convince ourselves how much truth there was in the boasted power of Motecusuma, of which so much had been said: we would gladly risk our lives in it, and therefore begged he would make preparations for this expedition. But, previous to commencing our march, we ought first to give some proof of our most humble submission to his majesty our emperor, by forwarding him a complete account of everything that had befallen us since our departure from Cuba. We also proposed that all the gold we had bartered for, and the presents sent by Motecusuma, should be forwarded to his majesty.

In answer to which Cortes said, that our ideas accorded exactly with his own, and that he had already spoken to the same effect to several of the cavaliers. There was merely one circumstance which caused him to hesitate, namely, that if each person took the portion of gold which fell to his share, too little would remain to be worthy of his majesty's acceptance. For this reason he commissioned Diego de Ordas and Francisco de Montejo, who were thorough men of business to see what they could make out of those men whom they might expect would demand their share. This was accordingly done, and they represented to every one that we were desirous of sending his majesty the emperor a present in gold, which, considering it was the first, ought indeed to be something valuable. In order, however, to make this possible, nothing remained but that each one should give up his share of the gold which had been made up to this moment. A great number of officers and soldiers had already signed their hands to that effect; yet every one was at liberty to act herein as he thought proper. Here was the paper, which every one who chose could put his hand to.

Every one, without exception, signed his name to the document, and agents were chosen to be despatched to Spain. These were Alonso Puertocarrero and Francisco de Montejo, to whom Cortes himself had already given above two thousand pesos. The best vessel of our squadron, furnished with the necessary provisions and manned with fifteen sailors, was selected to convey them. The charge of the vessel was given to two pilots, one of whom was Anton de Alaminos, from his being so well acquainted with the passage through the Bahama channel, and the first who had ventured that road. Upon this all of us, in common, drew up an account of our adventures expressly for his majesty, relating everything that had happened to us, and Cortes himself, as he assured us, likewise wrote a very circumstantial narrative, which, however, was not given us to read. The account was signed by all the authorities of the new town and ten soldiers, of which I myself was one. But there was likewise another account drawn up by all the officers and soldiers, the contents of which will be fully explained in the following chapter.

Fancisco de Montejo and Alonso Hernández Puertocarrero were among the men sent back on this ship on 26th of July, 1519. Both safely returned to Spain to petition the King on Cortés's behalf.

Our agents had a most favorable voyage to the Havannah, and thence through the Bahama roads: their further course was equally prosperous, and they very soon arrived at the Tercera isles, and from there to Sevilla, where they hired a carriage and posted to the imperial court residence, at that time in Valladolid. Here the archbishop Fonseca governed at will, he being, moreover, president of Indian affairs, and the emperor then still very young, and residing in Flanders.

...Upon this Puertocarrero, Montejo, Martin Cortes, the father of our general, the licentiate Nuñez, who was reporter to the royal council, and a near relation to Cortes, determined to despatch a courier of their own to the emperor in Flanders. They fortunately possessed duplicates of all our despatches and letters, as also a list of all the presents we had destined for his majesty.

With these papers they likewise sent a separate letter to the emperor with complaints against the archbishop and the whole of his doings with Diego Velasquez. In this resolution they were backed by other cavaliers who were at variance with the archbishop, who, upon the whole, had many enemies on account of his haughty behaviour and the abuse he made of the important offices he filled. And as the great services we had rendered to God and his majesty, in whom we had alone reposed our trust, were looked upon in a favorable light, it also happened that his majesty made the strictest inquiries into the whole affair. His majesty was so highly pleased with what we had done, that the dukes, marquisses, earls, and other cavaliers, for days together spoke of nothing but Cortes, our courageous behaviour, our conquests, and of the riches we had sent over. It was owing to this as well as to the unfaithful and distorted account which the archbishop had drawn up respecting these matters, and particularly because he had not sent all the presents, but kept the major part to himself, that he fell from that moment into his majesty's displeasure. In the meantime the archbishop's agents in Flanders had sent him information of all that had passed, which vexed him in no small degree, and if previously he had blackened Cortes and all of us to his majesty, he now boldly accused us of high treason. But the Lord very soon bridled his rage; for two years after he received his dismissal, and then in his turn experienced the curse of malice and contempt. We, on the contrary, were looked upon as loyal men who had rendered services to the crown, as shall be mentioned in the proper place. For the present the emperor informed our agents, that he would himself shortly visit Spain to investigate the matter more closely and reward us. Our agents, therefore, awaited his majesty's arrival in Spain.

Then he scuttled the rest of his ships and marched inland.

For more read the The Memoirs of the Conquistador Bernal Diaz. It's a very easy read. Specifically...

• Chapter XLII "How we elected Hernando Cortes captain-general and chief justice until we should receive the emperor's commands on this head; and what further happened"
• Chapter LIII "How we arrived in our town of Vera Cruz, and what happened there"
• Chapter LVI "How our agents passed through the Bahama channel with the most favorable wind, and arrived in Castile after a short passage; and of our success at court"
• Chapter LVIII "How we came to the resolution of marching to Mexico, and of destroying all our vessels, which was done with the sanction and by the advice of all Cortes' true adherents."