How do the Europeans suceed - Logistics its all Logistics the Europeans have to ship colonists faster than they die.
A typical ship of teh time was the Dutch fluyt (the Dutch had IIRC the biggest merchant marine in the 17th century) - with 25-30 men crew they could transport arond 100 passengers.
Assume that within 3-6 months around half of the first colonist die.
Assume a family size of 5 (maybe you will have to double for child mortality, but thats included in the 50% casualites

)
so one ships yields 10 men that can carry weapons.
Asume a colony needs at least 100 me to be able to defend itself you need 10 ships per settlement. Or rougly 100 ships to bring the settlement to 5000 inhabitants within 5 years (from then on a colony should sustain itself)
Overall I assume you need at least 10 such settlements to effectively control the Northen American East Coast - that is 1000 ships (200 per year) to take hold.
That would need the effort of several European nations working together - (almost an ASB thought) but maybe doable. After a 5 year intensive settlement programm you probably need less people going there and you can bring in more supplies with the same number of shipps wjhich wwill allow for faster growth of the colonies.
That assumes hostile indians, if you can "befriend" some tribes you might need less investment. But the South And Mesoamerican Civilisations will need MORE effort, so I assume you will be able to get a foothold in the north, and if lucky can expand after 100-200 years southward - but those organized states in the south will make it extremely difficult to dominate them...
And if you invest in the Americas during the 17 century you might lose Eastern Europe to the Ottomans...