(Male) primogeniture has never been the custom in Sweden, except for a FEW estates, mostly owned by the nobility, that were entailed - in Swedish, "fideikommiss". An entail can have any set of silly conditions, though if they are too silly they can be set aside by a court of law. An example today is the princely estate of Berleburg in Germany where the entail is - as is very common - left to the eldest son, but only if he marries an Aryan, noble, Protestant woman...since the current heir has a girl-friend who's half Mexican (i.e. not Aryan), Catholic and in no way noble, these clauses of the entail are today fought in German courts as being unreasonable/obsolete.
In Sweden entails were introduced in the 17th century and were most popular in the 18th century. In 1810 a law forbade the creation of new entails, but the old ones were allowed to continue until the law of 1964 which allowed the then current owner to continue as he was but the entail had to be broken when he (or she) died. At the time, there were about 200 entails.
I have no idea how many entails there were in 1810, but probably not in excess of let's say 250. So to say that this was a major factor in the emigration to America is quite incorrect. The nobility and landed gentry seldom had more than half-a-dozen children since they knew they had to provide for all of them, regardless of the entail. The eldest son (most commonly) did inherit most of the real estate which had to be handed down intact to the next heir (this is was entail means). But such an estate yielded a very nice income - which is not bound by entail conditions - and from this more, unentailed, property could be bought and given to other sons and also provide dowries for the daughters. The eldest son also had a moral duty to provide a home for his siblings, and assist them when they needed it (e.g., officer commissions and promotions were bought, not due to merit - this was the norm all over Europe). Also, the mother's money (people with entailed property always married money) could be used to provide for the younger siblings.
I'm sure you can find singular examples where someone of the nobility went to America because there was no future in Sweden, but you will find many, many more who went because of the adventure. But very, very few of the nobility or the educated classes emigrated at all - unless they were sent there to escape the consequences of a crime or a misdemeanour. It wasn't unheard of for young men to be sent to America because they had made a girl pregnant, e.g.