English equivalent proverb/idiom for this saying

I cannot think of any similar idiom that uses behaviours (e.g., pinching, rocking) to convey the idea that seems to be conveyed by the Tamil saying. However, Janus-faced, and equivalently but less poetically, two-faced, both seem to convey the idea of duplicitous, contradictory behaviour.

I'd paraphrase what your politician is doing as "creating a problem so that he can solve it". It's not an idiom though. So I googled it. And having typed "create a problem", google suggested an autocomplete:

create a problem sell a solution

Google reckons it has 290 million hits... but most of them are using words from the phrase not the phrase itself. By using quote marks to restrict it to sites including the words as two separate phrases, there are only 320 hits.

So it's not a widely used phrase, so you can't really call it an idiom. It's catchy though, and clear what it means. I might use it, or something similar, maybe:

Create a problem so that you can offer the solution

I found the various comments about the saying interesting and perceptive! But what is distinctive about this Tamil saying is that it invites a dual perspective: the meaning is to be gathered from taking into account both the 'old man' and the 'baby'. The truth is not only in the 'doing', but also in the 'who' it is done to.

On the one hand the old man pacifies the baby by rocking the cradle in order to both hide the fact that he has caused it pain, as well as to be praised as being caring. Throughout, the old man knows that the baby can never speak about its tormentor. He is malevolent and also diabolical. On the other hand, the 'baby' has no way of telling who has hurt it & how it happened because it cannot speak. It is helpless and also vulnerable in being deeply asleep. The baby feels fear and endures harm but cannot say who caused it.

The truth to be learned is somewhat of a conundrum: the action here encompasses a kind of inevitable harm.

The main difference is that this proverb seems to focus on causing a problem in order to solve it, whereas that one is more concerned with doing something cruel or unethical and then assuming a mantle of piety.

The Tamil saying shows both a malicious ruthless intention object & an utter helplessness in the face of such harm. The Hindi one is about the issue of 'hypocrisy'—the cruelty portrayed here is meant to highlight the 'hypocrisy': pretending to the world that is one very good while doing a lot of harmful things on the sly.

I think the idiomatic expression of playing Jekyll and Hyde seems the closest to creating a problem and appearing to resolve it. Here the attempt to reveal two opposite sides of a man’s personality. The tamil expression is more to do with playing the double game of satisfying both opposing sides.