As always, you don't repair things by randomly trying to test components - you need to fault find. As far as remote controls go common failures would be the ceramic resonator (often with a broken leg due to being dropped), the electrolytic capacitor (either O/C or broken leg again), or the IR LED (again drop damage, may be broken, may just be O/C).
The blue device is the ceramic resonator, and presumably 2.5MHz - make sure it's not broken, and check it with the scope, using a x10 probe (a x1 may well stop it anyway).
There's pretty well nothing else on there to test, nothing else ever really fails - but seriously check the pins on the blue ceramic resonator, that's a prime suspect for drop damage.