They only need to. But it makes little point for them to settle for that if they are able to get further.
The point is not that we should encourage them to go on, but that dismissing those who go on as not really refugees fail to acknowledge human nature.
If you've taken crazy risks to flee, it makes no sense to stop at the very first point where you're no longer at immediate risk of death if you e.g. believe you have a better chance to be allowed to stay in a different country, or if you believe you have better chances at a normal life if you go on. Whether or not those beliefs are right.
I did not suggest we should "just accept everyone". Just that the idea that continuing on is an indication of whether or not someone is genuinely a refugee is flawed.
A refugees who flees war only need to go to the closest safe area and they are away from the imidiate danger that make them refugees.
Those with most money are those who can afford moving the furthest away.
In most cases it's actually the men who travel all the way to ex Scandinavia and leave their wife and kids behind. They do this because they know they can get them reunited later on.
The people who really have no money are stuck in the nearest refugee camp. I certainly don't hold it against those with the means to try and make a better future for themselves far away from their homelands but that does not mean you just accept everyone.
Then you just end up like Sweden and frankly not even the Swedes are happy about that right now.