Remember that Jesus told the people they needed to repent (change their mind), because the kingdom of the heavens was near (Matthew 4:17) – they needed to change their mind about how they could enter the kingdom. They thought that if they simply obeyed the Law, they would gain entrance. In Matthew 5-7 Jesus explains that they needed perfection (5:48), and that they did not have it. So, they needed to be asking how they could gain that necessary perfection (ultimately it is only by faith in Jesus, as Jesus explains to Nicodemus).
As would become evident in Jesus’s teaching, having eternal life and entering the kingdom would refer to the same destiny, as the two overlapped (as in John 3:3, for example). The kingdom was a national promise, while eternal life is an individual one. A person had to have life in order to participate in the kingdom. So the question is whether the kingdom and heaven are synonymous. They are not, because ultimately the kingdom of the heavens comes to earth and does not remain in heaven. They are very connected, but are not synonymous.
There are also still national elements uniquely for the nation of Israel that are coming in fulfillment of covenant promises that the peoples or nations won’t necessarily have, though there will be people coming from east and west to dine in the kingdom (Matthew 8:11).
For us in the church age, we are citizens of the kingdom which hasn’t begun yet (hence, Colossians 3:1-4). We know that we will spend at least some time (i.e., the tribulation period of seven years), in heaven with Christ before He returns (with us) to launch His kingdom. Whereas the tribulation period is called Jacob’s Trouble (or distress) in Jeremiah 30:7, and in that context we are reminded that the seven year period is focused on the nation of Israel, and is a prelude to the coming of the Messiah’s kingdom. But those alive at the end of the tribulation who believe in Jesus will see their Messiah return to earth and will be with Him. So, heaven isn’t the issue for them. The issue is eternal life and entrance into the kingdom as it begins. This is one reason that believing in Jesus results in eternal life, not eternity in heaven. We will always be with Him, wherever He is (1 Thessalonians 4:17), and so we need to pay close attention to where He will be (see Revelation 21:3, 10, for example).
It is worth noting, for example, that in John’s gospel, for example, there is no connection at all with eternal life and going to heaven – this is because eternal life is about Who we are with, not where we are located.