I find these questions to be particularly easy to answer by finding a counterexample - think of the construction in a different context, and see if that construction makes sense.
So, let's try this in the context of...leg surgery!
In 1985, the doctors operated on my foot. As a result, I could walk again, and I chose to become a runner.
Success! This construction seems to work just fine in a different context as the past tense.
Now as to why you seemed to only read it as a hypothetical, I can't say for sure. I can guess however! Looking at the sentence:
As a result, I could be admitted to [a certain undergraduate school], and I chose to go there.
The thing that clarifies that this is in the past tense is the past tense use of "choose" at the end in, "I chose to go there." Maybe because the sentence is long-ish and busy before that point, you just weren't noticing the past tense use very clearly? Maybe you've heard "could be admitted" used as a hypothetical many more times than as past tense (which is likely to be the case) so your mind automatically thinks of it as a hypothetical? Maybe it's another reason. I really don't know.
Either way, both sentences are correct. Both require context, and both mean the same thing.
After thinking about it, I did find one difference between the two. I think your edited version makes a little bit more sense logically.
As a result, I was able to be admitted to [a certain undergraduate school], and I chose to go there.
In this case, you use "I was able to be admitted" - this implies that your friend not only could be admitted, but also that he WAS admitted, which makes more sense when you follow up with "I chose to go there."
"I could be admitted" doesn't have the implication that "I was able to be" carries with it, so the logical jump is bigger between "I could be admitted" and "I chose to go there."
However, I'll maintain that they are both correct, and this distinction is very nit-picky. In practice, both mean the same thing.