Capacitors are used for a variety of reasons. In some cases, as you mention, they are to shunt noise so that electromagnetic interference is lowered (that's the static on the TV thing). Motors in particular generate a lot of electrical noise, so capacitors help reduce this.
They are also used to keep the power level uniform - think of them as little springs or batteries that work 2 ways, like an energy bladder - too much energy they store up the excess, too little and they give stored energy back to the system.
While something may work with them removed, it won't be as originally designed, and more importantly tested for proper operation and longevity, as well as "playing nice with others".
What your describing as "melted plastic" is kind of exactly that - it's the potting compound used to keep the wires going to the innards of the capacitor separated (so they don't short out - one goes to power and the other to reference) and provide mechanical stability as well.
Michelle
(retired electrical engineer)