But that doesn’t answer the question of how Popes are validly elected, if there has never been a time-honored, consistently applied mechanism in place. Your statement also begs the very question what about when Popes aren’t even elected!
As stated before, Jesus didn’t tell Peter how to appoint a successor. However, Jesus DID give Peter (specifically) and the Church (in general - by telling all of the apostles after the Resurrection) the power to bind and loose. In other words, they were given the authority, guided by the Holy Spirit, to write rules that would govern the Church. These rules are what are now called “Canon Law”. BTW, the book of Canon Law is shorter than the Affordable Care Act, and most US laws today. Regardless, one of the provisions in Canon Law is how the successor to the Pope will be chosen.
The current Pope can actually change (or waive) any provision in Canon Law should he choose to do so, though usually Popes don’t actually put forth major changes without a synod of bishops discussing the proposed change. This is due to the fact that Canon Law is not unchangeable doctrine - it is made of changeable disciplines. Pope Francis could, technically, change the Canon Law to make it so that the next Pope is again hand-selected by the current reigning Pope (as it was originally done - Peter hand-picked Linus to be his successor). Or, he could decide that the Pope should be selected by lot (as is done in the Coptic Orthodox Church). But he probably wouldn’t make such a drastic change unless the synod of bishops signed off on it and told him it was a good idea.
The reason why the selection of the Pope is today so secretive and has many laws has to do with abuses, both secular and clerical, of the Papacy. Popes have been elected for a long time. But, as happens with many world elections, especially before secret ballots, many world leaders would coerce the electors (none-too-gently, either) to vote for a hand-picked candidate. The reason behind the conclave? Interregnums used to take years, not days, weeks, or months, due to the Cardinals refusing to vote for the Pope. The laity got fed up, forced them to meet, and locked them in their meeting place with a key (con clave). Eventually, the laity even took away their food and the comfort of a roof due to the Cardinals refusing to vote. Suffice to say, these tactics worked so well that they’re now encoded in Canon Law regarding Papal Elections.
But see, regardless of the man who is elected to be Pope, or the method upon which he obtained the Papacy, God gives him the ability to guide His Church without changing the faith or morals, just as He gave the Jewish High Priests the gift of prophecy regardless of how the High Priest became High Priest. As stated before, Caiaphas was hand-picked by the Romans to be the Jewish High Priest. For 800 years, the High Priesthood was purely hereditary, limited only to the proper Zadokite line (before this, it seems that it rotated amonst the Aaronite clans).
But, starting with the Seluecid empire (specifically with Antiochus IV), the ruling empire hand-selected the High Priests. When the Maccabees (in the Hasmonean line) were given the High Priesthood, it stayed within their family until the civil war between Astrilobus II and Hyrcanus II, after which Herod was given client kingship and started hand-selecting High Priests himself. When Rome took direct control of Judea, they hand-selected High Priests themselves. However, regardless of how the High Priest became High Priest, he was considered “God’s anointed”, and endowed with gifts from God, notably, the gift of prophecy.
And, it was considered a grave sin to curse the High Priest. This is why Paul retracted his curse and charge of hypocrisy against the High Priest (saying “God will smite you, you whitewashed wall”) after he was notified that the person he had just cursed was the High Priest.
If God could endow the Jewish High Priests with such gifts, regardless of how sinful they were or how they obtained the office, He could surely give the same gifts to the Vicars of Christ, regardless of how sinful they are or how they obtained the office, especially since Jesus, God the Son, personally guaranteed to Peter that He was giving him the keys to Heaven. That means the Pope has the authority to make decisions in Jesus’s name, guided by the Holy Spirit, just as a Chancellor/Vizier/Vicar/Prime Minister (same job, different title based on location and tradition) has the keys to the kingdom and wears the signet ring of the king (the Pope’s “ring of the fisherman”) and promugates laws and decrees in the name of the king, regardless of how he obtained the position.
So, to answer your main question - yes, Popes are Pope simply because they are the Bishop of Rome - the way men were validly selected to become Bishop of Rome changed over time, as secular/clerical abuses forced them to change. By the way, the Western schism was primarily caused by the French being upset at their loss of Papal control. As such, after the validly-elected Pope returned to Rome from Avignon, the French and their allies refused to acknowledge that he had been validly elected and had their own conclave. Of course this caused scandal, as the laity saw two Popes, not knowing which one was the “true Pope”. It was a sad state of affairs, but it (and the Avignon papacy that preceded it) is part of the reason why Cardinals are hesitant to elect Popes from the most powerful countries - the Pope needs to be independent of secular power.