Not a mechanic as such, but a background in Engineering and a long history of not wanting to pay someone for a job that invariably they just throw parts at!
It seems as though you are nearly there. CDI's rarely break down, it is much more likely something else. Some more things to try.
1. Has the bike got any failsafes, such as sidestand cut-off, clutch lever cutoff? Sometimes the microswitches stick and need to be freed off, especially if the bike has been sitting for a long time. Also, a previous owner may have bodged a bypass that may have come unbodged!
2. Check the ignition circuit for continuity to see if you have any breaks in the cabling.
3. Spray the switch gear with a good quality contact cleaner, not WD40, GT spray, etc.
Good luck and keep us updated. The weather is going to be good (in the UK) so it will great to get out on your bike.