Awesome job! I wrote up that sticky on the CDI swap, good job on the pics. I have had to explain many times that the contacts can be re-used in the new connector shell, provided they are in good shape. Your contacts look good.
As for the electric start, no its not related. I would almost bet that you have either a bad solenoid or a bad safety relay. On the back of the battery box along with the solenoid there is a small automotive type relay and the rectifier/regulator. The relay is energized by the neutral switch, it switches the ground for the neutral light and that safety relays coil. I have had a few now that the small safety relay was bad/burnt out. The inhibitor switch as the diagram calls it, has two Green/Red wires that are the input and output, its in series with the start button and the starter solenoid. Unplug it and jump across those two wires with a piece of wire, paperclip, whatever. Its a switched ground not positive so no fear of shorting it on something else.
If the starter solenoid now works after bypassing that relay then its just a bad safety relay. All this does is prevent running the starter while in gear, on mine i just bypassed it by pulling those two green/red wires out of the connector trimming them back and soldering together and tossed some heatshrink tubing over them and then tucked all back into place. You could replace the relay too if it really worries you.
If the starter solenoid still doesn't work then its most likely a bad solenoid.
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1984 Honda ATC200ES "Big Red"
1982 ATC200E "Hondie"
1988 TRX300FW "Project Quad" Still in progress....