First we need to narrow down the mode of the preposition "to", among others:
- directional indicator == towards
- infinitive indicator - e.g., he came to eat
- range indicator == till - e.g., from top to bottom
- fulfillment indicator - there are 2.64 litres to a US gallon.
- relationship indicator - who is she to you? My clock is synchronised to hers.
- He pointed the man to the door, when the man asked him for the exit.
- He pointed the gun to(wards) the door, when the man knocked.
- He pointed the gun to close the door.
Whereas, preposition "at" is a focus indicator. e.g.,
- He pointed the gun at the door.
- The world will end at noon.
Let us compare the modes:
- She threw the ball to him. (She meant to give the ball to him)
She threw the ball at him. (She did not mean to give him the ball but to use it as a projectile onto him).
He directed the man to the door.
He directed the gun at the door.
He redirected his aide to close the door.
- He redirected his butt to close the door.
Therefore, we should not place the cart before the horse. We don't decide the differences between styles of chopsticks but discover the food we would be eating to use the appropriate style of chopsticks.
In applying Mathematics to Engineering, rather than asking repeatedly the difference between Sine, Cosine and Tangent, we discover the available parameters and the required results, and then look into the available features to then decide which provides features that can be most effectively deployed.
Similarly, here, we should first discover the mode of indirection we wish to project and then see which of the modes are available for us to apply appropriately.
There is not a singular definition of "to" but a spectrum of modes (not disregarding usage as an adverb). Depending on the mode you intend, you may use one of the modes of "to" or "at", and there may be overlap in the various modes across the spectrum of the two prepositions.