What would happen if a run start capacitor is installed across the 220 VAC?

noone has said this is a good idea.

run/start capacitors typically have motor windings connected in series with them which limit the damage if the capacitor shorts out.

For connection across the power lines a "class X2" capacitor is indicated.

Also filter X capacitors are designed to pass radio frequency, run (or start) capacitors may not be. thus you may see adequate performance with a much smaller capacitor that the motor caps.

larger X2 caps can be had in the form of PFC capacitors.