The basic problem with this episode is that King Solomon's Mines and the Ark of the Covenant are grounded mostly in mythology. From the time of the First Temple in Jerusalem which is supposed to incorporate the reigns of King David and King Solomon nearly 3000 years ago, we have very little surviving evidence. The Temple was conquered and destroyed by the Babylonians in 605 BCE. Yes there probably was an original Temple (not to be confused with the Second Temple which dates to the time of Jesus). The trouble is we don't even have physical evidence of King David or King Solomon aside from what is written in the Jewish/Hebrew Bible, aka The Old Testament. So trying to find King Solomon's Mines seems like a stretch.
About as worse is the so-called Ark of the Covenant which supposedly housed the 10 Commandments in the "Holy of Holies" in the First Temple. It became particularly famous when the film "Raiders of the Lost Ark" was released in 1981. Indiana Jones and the diabolical Belloq hunt for the Ark in the Middle East. If it had existed, the Ark would most certainly have been lost or destroyed when the Babylonians conquered Israel, again in 605 BCE. Probably a 99.99% chance it was destroyed rather than survived assuming it had even existed. Interestingly, the earliest Jewish writing survives from around the time of the conquest which makes it problematic whether the Old Testament can be relied on concerning the Ark's original existence. It could be a mythology in the same way the Holy Grail is largely a mythological artifact.
Several "if's" make its survival very problematic. If the Ark did exist, it would have been the most highly-prized treasure in all of Judaism, particularly in the First Temple in Israel. The idea that it had been transported before the Babylonian conquest seems downright absurd. It would be like giving the French our original manuscript of the Declaration of Independence to the French for safe-keeping. More than likely, it was destroyed like everything else. Anything made of gold would have been melted down and reused by the Babylonians or any other conquerors. That's what conquerors did in Antiquity.
Of course the nuttiest idea has to be that the Christian Church in Ethiopia owns the Ark of the Covenant in a special secluded church. Ethiopia is one of the poorest countries in the world, and the idea that only a special "keeper" can view it has to mean they don't have it. (A former British army officer and scholar of Ethiopia claims he saw it in the 1940's. He said what he saw looked like a plain medieval wooden box, not an ancient ornate chest.) For one thing, if they had it on display then they could incite tourism and make money from travelers and pilgrims! However it would also mean the Ark could be scrutinized. They must know that whatever they have isn't the real deal, or else they'd be showing it. It's easier to let the "mystery" persist. But that's not going to entice any kind of tourism.
The documentary furthers that the Ark could have gone elsewhere. Yes and maybe if we keep looking we could find the silver dollar George Washington threw across the Potomac. It's all very speculative. Most scholars won't rule out that there had been an Ark but it's an iffy proposition. How much evidence for either of these things, the Ark and the Mines? Almost none.