To trust in God was a major element of the Musar yeshivas of Navardok. But it is implicit in almost all Litvak yeshivas. Where I went was the Mir in NY and before that was Shar Yashuv. But in both places there was a basic idea that all you need to do is to learn Torah and God will take care of the rest.

And trust in God had other benefits. It is brought in the Sefer HaMidot of Rav Nahman that by trust in God good thoughts are drawn to you. That is it is a shield against evil thoughts and evil world views.

In spite of this, in some places Torah is used to make money. But that is not the idea behind a Litvak yeshiva in its original sense.

But the point is that to learn to trust in God depends a lot I think on being in a Litvak yeshiva where that is the prevailing world view --that trust in God is the main thing in life. By being with like minded people one can gain that kind of world view himself.

In Israel I have very little familiarity with the situation, but the best and are Ponovitch and Brisk.

In terms of the issue whether all one is supposed to do is to learn Gemara I have found it helpful to depend on the opinions of Ibn Pakuda and other Rishonim that include Physics and Metaphysics in the category of learning Torah.