How can AIC converge in the limit when even 2 parameter models can have infinite VC dimension?
I tried to find a simple and accessible analysis of AIC. A definitive work seems to be Barron, Birgé, Massart, "Risk bounds for model selection via penalization"
https://link.springer.com/article/10.1007/s004400050210
though I can't vouch for accessibility.

Instead, let's analyze your sinusoids example. Suppose I have a parametric class of densities over $[0,1]$, parametrized by a single parameter $\alpha$, which is the frequency of the sinusoid. I take the positive part of the sinusoid and normalize the function to integrate to $1$ -- now it's a density. I can choose the $\alpha$ via any of the model selection techniques, including maximum likelihood and AIC. For any fixed $\alpha$, I think all of these will converge (maybe even almost surely) to the true $\alpha$. So why doesn't this contradict the fact that this family of functions (when thresholded at $0$) has infinite VC-dimension?

It's all about the order of quantifiers. Finite VC-dimension means that you can specify a sample size $m_0(\epsilon,\delta)$ that will suffice to achieve $\epsilon$-accuracy at $\delta$-confidence -- in advance of seeing any data. In our model selection example above, you will not be able to specify such an $m_0(\epsilon,\delta)$.

There are certainly infinite-VC classes that are learnable in this sense, which is weaker than PAC. For any target concept and distribution on the integers, you will eventually see enough examples to know the true concept to any desired accuracy -- you will just not be able to specify this sample size independently of the distribution and the target concept. (See Ex. 0.2(c) here
https://www.cs.bgu.ac.il/~asml162/wiki.files/hwk0.pdf
).