Birthday Problem: Hypothetically Not Using the Complement My Professor said that the probability that at least 2 of $k$ people have the same birthday if we didn’t use the complement is the sum of the probabilities that exactly 2 have the same birthday, plus the probability 3 have the same birthday, and so on…
However, I wanted to verify this and believe I thought of a case where this is not true: the case where exactly 3 and exactly 4 have the same birthday. I.e. the event that exactly 2 and exactly 3 people have the same birthday are not disjoint. therefore, we need a different method to calculate the probability not using the complement. Is this valid?

If you have a random variable $X$ (e.g., the maximum number of people in the sample that share a birthday) that can only take on values $0,1,2,...,k$ then it is certainly true that:
$$\{ X \geqslant 2 \} = \bigcup_{x=2}^k \{ X = x \}.$$
Consequently, you then have:
$$\mathbb{P}(X \geqslant 2) = \sum_{x=2}^k \mathbb{P}(X = x).$$
Presumaby this is what your professor is talking about in this case.  Your misgivings seem to arise from ambiguity about the event whose probability you are computing.  If you can clarify that with your professor then this might become clearer.
