I will tell you simply why, simply saudi arabia or any rich arab gulf state don't accept refugees , for a syrian to go to saudi arabia , he would have to pass jordan, getting into jordan means automatically getting into the large zaatari refugee camp (which is a nightmare to live in), or getting shot at the boarder (depending on your luck) , even many syrians who go from syria to turkey get shot at the boarders.. , let's assume a person makes it into jordan illegally , and wants to go to saudi arabia , the boarders are well guarded between jordan and saudi arabia , and even if the person makes it into saudi arabia, he might risk deportation back to jordan.. or maybe get shot at the boarders , or even die in the vast desert .
i might note that even getting to the jordanian or turkish boarders might be impossible within syria since, well, there's a war going on there.
Trust me, even syrians are asking the same questions like "why the rich arab countries won't take us" , "we speak the same language and share the same religion (not all syrians are muslims but most of them)" .
They choose Europe because they know that they will live in a good condition, they know that they won't get killed (well maybe in the sea , but after arriving to greece there is little to no risk)
the main countries the refugees go to in europe are : the Netherlands , Germany and Sweden . with each person choosing a country depending on the laws of that country (for example: the laws in germany makes it easier for a person to get his family "legally" out of syria and into germany, and that's why germany is the number 1 destination)
There is also the debate of "why are syrians leaving turkey which is safe", it is safe in the sense that you won't get killed , but it's difficult to get a work permit , so unless you are full of money and can open a business, you will suffer in turkey, while in europe , they are treated equally after getting integrated in the society.. so yes ... they are not just refugees , but also economic migrants with really really bad conditions.
Long story short , the logic of this migration is :"i am a syrian, syria is too dangerous to live in , and i won't be able to provide for my family, there is no future ,in turkey i don't have a future, i will keep living on aids , it's impossible to work, i go to europe, get integrated in the system, i can work, i have equal rights, i feel safe, they won't treat me like an animal"
I understand that it's not "europe's responsibility" , i personally blame the foreign policy of the US and UK when they invaded iraq and destabilized the country , which led to the creation of "the islamic state in iraq" , which then expanded to syria hence (the islamic state of iraq and syria , ISIS)
No syrian thought of taking this journey before the war..