Why are ALL Tehillim permitted to be read on Shabbat?

Question

I’m curious why it is permitted to read/recite Tehillim “carte blanche” on Shabbat. By that I mean, why are ALL of the Tehillim acceptable? As we know, on Shabbat we are not to pray for our needs, for what we lack. Indeed the Yerushalmi forbids davening for needs on Shabbat. Shabbat is a day of happiness. Davening for our needs highlights the pain or at least what is missing and desired in the davener’s life. Our teaching is that that is the reason we remove the middle thirteen berachot of Shemoneh Esrei on Shabbat. So why are ALL the Tehillim permitted and not just the Tehillim that praise and/or thank and/or exalt HaShem? So many Tehillim do indeed express our needs and call out for succor and relief. Any thoughts would be greatly appreciated. Shavua Tov. Shanah Tovah!