Mark 13 is not critical to dating this gospel, but can help corroborate external evidence, and perhaps help improve our precision in dating it.
The earliest external evidence we have, from a second century bishop named Papias, says Mark was based on Peter's preaching:
Mark having become the interpreter of Peter, wrote down accurately whatsoever he remembered. It was not, however, in exact order that he related the sayings or deeds of Christ. For he neither heard the Lord nor accompanied Him. But afterwards, as I said, he accompanied Peter, who accommodated his instructions to the necessities [of his hearers], but with no intention of giving a regular narrative of the Lord's sayings. Wherefore Mark made no mistake in thus writing some things as he remembered them. For of one thing he took especial care, not to omit anything he had heard, and not to put anything fictitious into the statements.
Irenaeus, writing in the late 2nd century, explained how the written gospels were a continuation of the apostles' teaching:
Matthew also issued a written Gospel among the Hebrews in their own dialect, while Peter
and Paul were preaching at Rome, and laying the foundations of the Church. After their departure, Mark, the disciple and interpreter of Peter, did also hand down to us in writing what had been preached by Peter. Luke also, the companion of Paul, recorded in a book the Gospel preached by him. Afterwards, John, the disciple of the Lord, who also had leaned upon His breast, did himself publish a Gospel during his residence at Ephesus in Asia.
(Irenaeus seems convinced that Matthew was the first gospel to be written, which is problematic for modern scholars, who agree that Mark was written first. But he also says this gospel was written in Hebrew, which would not be the Greek gospel of Matthew we have in the New Testament.)
However, if Papias' and Irenaeus' statements about Mark are correct, that Mark wrote it all down after the deaths of Peter and Paul, then we have a firm lower limit of around 65 AD.
Beyond this, we don't have many external clues to the dating of Mark's gospel, other than the fact that material from it is quoted verbatim—and sometimes expanded—in Matthew and Luke. This has led most scholars to conclude that Mark was written first.
However, we can work backwards to get an estimate of the latest possible date for Mark.
A letter written by Clement, Bishop of Rome, quotes extensively from the gospel of Matthew. If we can estimate the date of 1 Clement, we can establish an upper limit for Matthew.
Clement opens with a reference to:
the sudden and repeated calamities and reverses which are befalling us
He later refers to Christians who had been persecuted in the past, including Peter and Paul. If Paul and Peter died during the Neronian persecution (64-68 AD), then the "sudden and repeated calamities" of 1 Clement belong to a later persecution, most likely that by Domitian around 95-96 AD. (The book of Revelation was also likely written during this time.)
If 1 Clement was written in 96 AD, his quoting from the gospel of Matthew implies that Matthew was written prior to that. If Mark was written before Matthew, then we have a firm upper limit for when it could have been written.
Most scholars slice at least 10 years off that upper limit, to give the gospel enough time to be copied and circulated to Palestine, where Matthew's gospel was written, and then for Matthew's gospel to be copied and circulate to Rome, where Clement quoted from it. So the external evidence points to somewhere between 65 and 85 for Mark's gospel.
The only event mentioned in Mark which happened between 65 and 85 is the destruction of the temple. Jesus refers to it as a future event:
Do you see these great buildings? Not one stone will be left here upon another; all will be thrown down. —Mark 13:2
But the question is, was it still a future event for Mark? The gospel writer adds a parenthetical remark in verse 14 that indicates he knows what Jesus is talking about, and expects the readers also to be familiar with it:
But when you see the desolating sacrilege set up where it ought not to be (let the reader understand), then those in Judea must flee to the mountains… —Mark 13:14
Luke, in his version, paraphrases this as:
When you see Jerusalem surrounded by armies, then know that its desolation has come near. Then those in Judea must flee to the mountains… —Luke 21:20-21a
This happened during the siege of Jerusalem in 70 AD. So the "desolating sacrilege" was likely understood by the gospel writers as a reference to this event.
Mark expected his readers either to know the siege as a current or recent event, or to know that it was imminent. The war between Judea and Rome lasted from 66 to 73 AD. It is likely, based on Mark 13:14, that this gospel was written during the war or shortly after it.