"the whole multitude of men living at the same time" is the applicable definition in the greatest number of NT verses.
This proves nothing. It's not the only definition. Look at this passage..
Matthew 23:33 Ye serpents, ye generation of vipers, how can ye escape the damnation of hell? 34 Wherefore, behold, I send unto you prophets, and wise men, and scribes: and some of them ye shall kill and crucify; and some of them shall ye scourge in your synagogues, and persecute them from city to city: 35
That upon you may come all the righteous blood shed upon the earth, from the blood of righteous Abel unto the blood of Zacharias son of Barachias, whom ye slew between the temple and the altar. 36 Verily I say unto you, All these things shall come upon this generation.
The word "generation" in verse 33 is a different word than "genea", so, just to be clear, I'm not making that reference part of the point I'm going to make regarding the use of the word "genea" in verse 36. I include verse 33 to show that Jesus is talking about a type of people here that he calls "vipers". And, He, of course, also called them hypocrites several times in Matthew 23. And they were murderers of innocent servants of God.
In verse 36, "this generation" includes those vipers and hypocrite Pharisees and scribes that Jesus was harshly criticizing, but look at verses 35 and 36. Jesus indicated that "this generation" was responsible for all the righteous blood shed on the earth going all the way back to Abel and included "Zacharias son of Barachias" (Zechariah son of Berekiah in other translations), which may refer to the father of John the Baptist, but it doesn't really matter. The point is that "this generation" even includes Cain since he killed Abel. It also included all who killed any righteous people like Abel from that point up until Jesus was speaking in Matthew 23. Think about that. This means that Jesus indicated that "this generation" was not just those living at that time, but rather was a reference to a certain type of people that Jesus has in mind here in this passage. Vipers. Hypocrites. Murderers of innocent people like Cain.
So, again, seeing that "this generation" even included Cain and all who killed the righteous since Cain, you should see that "this generation" is a reference to a type of people and not just those who were living at that time. Some of those in "this generation" who were killed in 70 AD were not even alive yet when Jesus gave His Olivet Discourse. So, to claim that "this generation" only referred to people that were alive when Jesus was speaking is not true. So, "this generation" does not refer only to those who were alive at the time in either Matthew 23:36 nor in Matthew 24:34. Instead, in each case, "this generation" refers to a type of people, including Cain, and including some who were not born yet when Jesus was speaking, but would be among "this generation" that was killed in 70 AD.
With that said, "this generation" in Matthew 24:34 has a different context from "this generation" in Matthew 23:36. It again refers to a certain type of people, but not just those who were vipers, hypocrites and murderers of innocent people. Those people didn't literally all pass away in 70 AD, but in Matthew 24:34, I believe Jesus was talking about all of "this generation" passing away. He was talking about the literal, complete passing away of "this generation", in other words. Which, in relation to the Pharisees and scribes and unbelieving Jews in general, did not occur in 70 AD as some survived and were taken captive.
So, in the case of Matthew 24:34, it refers to the Jews in general passing away, in my opinion. The word "genea" can refer to a type of people, like it does in Matthew 23:36, but also can refer to a race of people. Notice in the very next verse after Jesus talks about "this generation" passing away He refers to heaven and earth passing away (Matthew 24:35). To me, that shows that what He had in mind in relation to the timing of "this generation" passing away was that it would occur when heaven and earth pass away at the second coming of Christ at the end of the age.