Forget about the capacitors for now, that board has been powered for a good while with some ionic liquid on it (cat pee is common under TVs). The copper in many of the vias (and maybe some top side tracks in the corner) is corroded out.
Clean locally, first with a mixture of distilled water and IPA, several times (IPA alone won't dissolve the salts) and then finally neat IPA to remove the moisture.
You will need to repair each of the rotted via holes by feeding bare tinned copper wire through them (hopefully standard 22 AWG), scraping away solder resist from the tracks until you reach clean copper, and then solder on both sides.
If the corrosion hasn't got under the BGA packages, you may stand a chance. That square IC package (35240?) next to the regulator looks bad though, clean around it carefully and make sure none of the pins have corroded away.
P.S. Have you looked at the underside of the board?