I read Shmueli's "To Explain or to Predict" (2010) a couple of years ago for the first time and it was one of the most important readings for me. Several great doubts come to solve after such reading.
It seems me that the contradictions you notice are less relevant that it seems to be. I try to reply to your two questions together.
My main argument is that your point 3 do not appear at pag 307 (here there are the detail) but at the begin of the discussion – bias-variance tradeoff argument (par 1.5; in particular end of pag 293). Your point 3 is the core message of the article. (See EDIT)
Your points 1 and 2 are related to the sub-argument of model selection. At this stage the main important practical difference between explanatory and predictive models do not appear. The analysis of the predictive models must involve out of sample data, in explanatory models it is not the case.
In predictive framework, firstly we have model estimation, then model selection that is something like evaluate the model (hyper)parameters tuning; at the end we have model evaluation on new data.
In explanatory framework, model estimation/selection/evaluation are much less distinguishable. In this framework theorethical consideration seems me much more important that the detailed distinction between BIC and AIC.
In Shmueli (2010) the concept of true model is intended as theoretical summary that imply substantial causal meaning. Causal inference is the goal. [For example you can read: “proper explanatory model selection is performed in a constrained manner … A researcher might choose to retain a causal covariate which has a strong theoretical justification even if is statistically insignificant.” Pag 300]
Now, the role of true model in causal inference debate is of my great interest and represent the core of several question that I opened on this web-community. For example you can read:
Regression and causality in econometrics
Structural equation and causal model in economics
Causality: Structural Causal Model and DAG
Today my guess is that the usual concept of true model is too simplistic for carried out exhaustive causal inference. At the best we can interpret it as very particular type of Pearl’s Structural Causal Model.
I know that, under some condition, BIC method permit us to select the true model. However the story that is behind this result sound me as too poor for exhaustive causal inference.
Finally the distinction between AIC and BIC seems me not so important and, most important, it does not affect the main point of the article (your 3).
To be clearer. The main message of the article is that explanation and prediction are different things. Prediction and explanation (causation) are different goal that involve different tools. Conflation between them without understood the difference is a big problem.
Bias-variance tradeoff is the main theoretical point that justify the necessity of the distinction between prediction and explanation. In this sense your point 3 is the core of the article.
In my opinion the fact here is that the problems addressed by this article are too wide and complex. Then, more than as usual, concepts like contradiction and/or paradox should be contextualized. For some readers that reads your question but not the article can seems that the article at all, or at least in most part, should be refuse, until somebody do not resolve the contradiction. My point is that this is not the case.
Suffice to say that the author could simply skip model selection details and the core message could remain the same, definitely. In fact the core of the article is not about the best strategy to achieve good prediction (or explanation) model, but to show that prediction and explanation are different goal that imply different method. In this sense your point 1 and 2 are minor and this fact resolve the contradiction (in the sense above).
At the other side remain the fact that AIC bring us to prefer long rather then short regression and this fact contradicts the argument at your point 3 is refer to. In this sense the paradox and or contradiction remain.
Maybe the paradox come from the fact that the argument behind point 3, bias-variance trade-off, is valid in finite sample data; in small sample can be substantial. In case of infinitely large sample, estimation error of parameter disappear, but possible bias term no, then the true model (in empirical sense) become the best also in the sense of expected prediction error.
Now the good prediction properties of AIC is achieved only asymptotically, in small sample it can select models that have too many parameters then overfitting can appear. In case like this is hard to say precisely in what way the sample size matters.
However in order to face the problem of small sample a modified version of AIC was developed. See here: https://en.wikipedia.org/wiki/Akaike_information_criterion#Modification_for_small_sample_size
I done some calculus as examples and if these are free of mistake:
for the case of 2 parameters (as the case in Shmueli example) if we have less than 8 obs AIC penalizes more than BIC (as you says). If we have more than 8 but less than 14 obs AICc penalizes more than BIC. If we have 14 or more obs BIC is again the more penalizer
for the case of 5 parameters, if we have less than 8 obs AIC penalizes more than BIC (as you says). If we have more than 8 but less than 19 obs AICc penalizes more than BIC. If we have 19 or more obs BIC is again the more penalizer
for the case of 10 parameters, if we have less than 8 obs AIC penalizes more than BIC (as you says). If we have more than 8 but less than 28 obs AICc penalizes more than BIC. If we have 28 or more obs BIC is again the more penalizer.
Finally let me remark that if we remain very close to author words we can read that she do not explicitly suggest to use AIC in prediction and BIC in explanation (as reported at your point 1). She essentially said that: in explanatory model theoretical consideration are relevant and in prediction no. This is the core of the difference between these two kind of model selection. Then AIC is just presented as “popular metric” and its popularity come from the idea behind it. We can read: “A popular predictive metric is the in-sample Akaike Information Criterion (AIC). Akaike derived the AIC from a predictive viewpoint, where the model is not intended to accurately infer the “true distribution,” but rather to predict future data as accurately as possible”.