I am trying to understand the physics behind a single sailed boat. I have read the following questions

Sailing against the wind. Is this a fair model?

What exactly is the 'lift' of a sailboat as explained by Bernoulli principle

I've also been following this explanation which seemed to be helping but I am still confused. Full disclosure: my physics is pretty weak and I've never sailed, but my maths should be sound.

I am (now) aware of the two components that provide the force that moves the boat: The "Newtonian aspect", where the wind collides with the sail and this produces a force and the "Bernoulli aspect" resulting from pressure differences resulting from the air passing around the sail.

(If people think this would be better as two questions, happy to break it down, but I suspect one or both will be silly)

Newtonian Aspect

I understand this in the most simplest sense, a sail perpendicular to the wind will produce a force on the boat proportional to the relative wind speed. As the sail is angled in to the wind, the force becomes proportional to the Sail normal component of the wind vector.

In the sailing explanation, it implies that the Newtonian aspect can be used to explain going into the wind. This is also hinted at in the first question. However, I don't understand how this can be. Surely the force applied to the sail, and hence the ship, is always in the direction of the wind. Hence it can never provide a force into the wind. The article seems to suggest you can get away without the Bernoulli effect, but I don't see how.

Bernoulli Aspect

This is the area I am even more lost with. My understanding is that somehow, you can angle the sail in such a way that the air that goes past on one side knows to go faster which reduces the pressure, giving a force. However, I cannot seem to connect that intuition to any kind of applicable formula. Specifically, what determines the direction of the force and what is it proportional to?

I'm trying to simulate a boat for a game and any help on which formulae I actually need to be plugging in would be hugely appreciated :)