Hi TMK,
I apologize for the delay in offering a response.
Yes, there are plenty of anecdotes from both Scripture and secular history that indicate an importance of Mt. Sinai even after the events in the books of Exodus, Leviticus, Numbers and Deuteronomy.
In Judges, we read the Song of Deborah in which the LORD is referred to as "This Sinai" or "the One of Sinai" (Judges 5:5).
In Psalm 68, David referred to Sinai twice as being where the LORD dwells.
ELIJAH & SINAI
In fact, after reading your question, I immediately thought about Elijah. I'm referring to what happened after the demonstration of God's power on Mount Carmel.
In I Kings 18, we read the account of how Elijah challenged the prophets of Baal. He told the prophets of Baal to make a sacrifice on an altar. Elijah, in turn, repaired the altar to the Lord that had been torn down and subsequently prepared his sacrifice upon it. Elijah and the prophets of Baal had agreed that the God that answered by fire would be God (I Kings 18:24).
After God consumed Elijah's sacrifice on this altar, the people agreed that "The LORD, he is God. The LORD, he is God" (I Kings 18:39).
Afterward, the prophets of Baal were captured, taken down the mountain and slain by the sword in the Kishon Valley. Elijah subsequently prayed for rain...and it rained.
Yet, the fact that Elijah had all of the prophets of the Baal slain angered King Ahab's wife, Jezebel. She was intent on killing Elijah.
What did Elijah do?
He fled for his life!
Where did he go?
After a stop in Beersheeba, he fled to Mt. Sinai.
Well, textually speaking, Elijah went to "Horeb, the Mountain of God" (I Kings 19:8). In the Scriptures, we find that the names "Sinai" and "Horeb" are interchangeable -- possibly just two names for the same mountain or one being the range and the other the summit. Either way, Moses is referred to having met God on Horeb and Sinai.
According to I Kings 19, Elijah was fed twice by an angel of the Lord, because he needed his strength. After all, the journey of Elijah would take "40 days and 40 nights."
It was in that cave on Horeb that the Lord asked him, "What are you doing here, Elijah?" Elijah was told to stand up on the mountain to stand before the Lord (verse 11). After a wind, fire and earthquake (all of which the Lord was not in), Elijah heard that "still small voice" (verse 13). Elijah covered his head and stood at the mouth of the cave.
We then read that Elijah stated his qualms by saying that he felt all alone. He went on to say that the Israelites -- the people that God had chosen -- had forsaken God's covenant (made at this very mountain), torn down the altars to God and now wanted to kill Elijah.
God encouraged Elijah by telling him that there were 7,000 faithful in the land of Israel. So, God sent Elijah to the Desert of Damascus to anoint Hazael as king of Aram, Jehu as king of Israel and Elisha to succeed him as prophet.
Elijah lived nearly hundreds of years AFTER the Exodus. Yet, at that time, the Bible records Horeb (Sinai) as the "Mountain of God" -- and somewhere that Elijah fled to during his time of despair.
WHEN WAS IT LOST?
I'm not sure when/where in the timeline of history it was that the significance of Mt. Sinai faded or was lost (in terms of its actual location).
Obviously, the mountain existed outside of the Promised Land. That is, Sinai was not in the land that God promised to Abraham's descendants. Aside from Elijah, there are no biblical accounts of pilgrimages to Mt. Sinai.
TRADITIONS OF MEN?
I suppose that it was a good thing that the Israelites did not make pilgrimages to Sinai. After all, the Samaritans began as a sect that saw Mt. Gerizim as a new "holy" mountain where they were to go for worship. After all, it was the mountain selected for the sacrifices of blessings once the Israelites crossed the Jordan into the Promised Land (Deuteronomy 11:29; 27:12; Joshua 8:33).
The Samaritans believed the notion that this mountain was holy and chosen by God for worship rather than the Temple in Jerusalem. It's as if they took a place from Biblical antiquity and turned it into a venerated relic or mandated "holy" place. They forget that the Ark and Tabernacle were not permanently posted on Mt. Gerizim. I suppose that this could have been the case in regard to Sinai.
THE APOSTLE PAUL'S ANALOGY
In Galatians 4, Paul writes an analogy that compares Hagar and Sarah. In verse 25, he describes Sinai as being in "Arabia" (which is generally regarded as east and south of the Dead Sea and Negev) -- which differs from the idea that it existed in what is now known as the Sinai Peninsula (more on that later).
Paul mentioned Abraham's two sons -- Ishmael and Isaac. One was born by the flesh and the other by a promise. He compared Hagar the slave woman with Sinai and, subsequently, Jerusalem (and slavery to the Law). Paul goes on to describe Sarah as the "Jerusalem above" -- and that we, like Isaac, are children of promise (verse 26-28, 31).
Paul said this to indicate that we are not children of slaves -- the law (represented by Paul as Haggar, Sinai and Jerusalem). We are children of the free woman -- represented as Sarah and "Jerusalem which is above" (verse 26).
Thus, for us, we aren't bound to ritualistic pilgrimages or even adherence to the Law of Sinai or worship in Jerusalem. As Jesus told the Samaritan woman at the well, true followers would worship neither on a mountain (in the Samaritans case, Mt. Gerizim) or in Jerusalem but in "spirit and in truth" (John 4:21-24).
WHERE IS SINAI?
Interestingly, Paul the Apostle (in the 1st Century) mentioned in Galatians 4 that Sinai was in Arabia. This would coincide with the notion that it was east of the Negev and within range of Midian (which was located east of the Gulf of Aqaba). Moses was the son-in-law of Jethro, a priest of Midian. He was tending Jethro's flocks when he encountered the burning bush at the wilderness of Sinai (Exodus 3:1-3; Acts 7:30).
So, at that point of time, the location of Sinai seemed to have been known by Paul -- a learned Pharisee of Pharisees. It's possible that Paul was stating this from his knowledge as a Pharisee who was also the son of a Pharisee and/or by specific divine inspiration from the Holy Spirit. We simply know that Paul stated a location.
There have been some individuals who argue that "Arabia" (as mentioned by Paul) encompassed more than what we consider "Arabia" later in history. However, what is now known as the "Sinai Peninsula" was not inhabited by "Arabians" at the time of Paul either.
Josephus, the 1st Century Jewish-Roman historian, also seemed to indicate a location for Sinai too. However, Josephus only mentions that Sinai was "highest of all the mountains that are in that countrey." However, he also mentioned that the Sinai was filled with jagged rocks and had a high altitude. Interestingly, Josephus also claims that it took the Israelites three months to arrive to Sinai after the were "removed from Egypt."
https://penelope.uchicago.edu/josephus/ant-3.html#EndNote_Ant_3.9a
Even decades before Josephus was born, Philo of Alexandria mentioned Sinai in a similar manner. Yet again, Sinai was described as the highest mountain in that district and very difficult to climb. This seems to rule out relatively easy climbs like Har Karkom in the Sinai Peninsula. Even the Mt. Catherine location seems puzzling because it doesn't fit this description very well (given how inaccessible the base of the mountain is too).
MOUNT SINAI...IN THE SINAI PENINSULA?
There is a reason why the Sinai Peninsula is called the "Sinai Peninsula." It goes back to either Egeria (a woman who made a pilgrimage to the "Holy Land" in the 4th Century) or, more likely, Queen Helena, the mother of Emperor Constantine in the 3rd Century.
Helena began designating many locations and building shrines at places suspected to be locations from the Scriptures. However, she had never even visited those areas. Many of these locations were subsequently turned into "holy" sites with monasteries and churches built on the premises. Unfortunately, there is little evidence as to why those sites were selected. However, given her influence (her son was the ruler of the Roman Empire), the places stuck and became Catholic traditions.
Still, these sites were selected in the first third and fourth centuries A.D. for some reason.
