No, wrapping the wire around is not a solid electrical connection - it's a good mechanical connection, but you need to solder the wrapped wire for maximum current transfer and permanence.
Did you try briefly grounding the trigger tab of the coil to see if it fires? That will tell you A) that the coil works B) that your "power side" supplying the coils is sufficient - thereby splitting the circuit in half of what needs to be troubleshooted.
http://stevekulpa.net/pinball/bally_flipper1.htm