Flipper solenoids are way overloaded for the first 100ms or so. To be strong enough to kick steel balls around, they have to be. So you need some way to turn the current down after the initial kick. Today this usually involves turning down the duty cycle for the coil power after the first 100ms or so. Older setups with dual coils and switches are mostly for repair work.

There are lots of good solenoid driver ICs around, but they're mostly sized for automotive applications. Pinball flippers typically need 5A at 50V, which is a lot to ask of an IC but no problem with an external power MOSFET. There are pinball boards for this. (http://www.allteksystems.com/products-mpu-replacements.html#...) No need to use electromechanical relays and put up with burned contacts any more.

(I'm interested in this problem because I restore old Teletype machines, which have similar electromagnet drive problems. The classical solutions are huge and inefficient by modern standards.)