The last time that happened to me - the battery was kerplunkt. It would measure with some reasonable voltage when resting, but as soon as a load was put on - even just the idiot lights when switching the key on, the tach would swing upwards and then drop back down, and everything would be pretty much dim/dead. No starter action, except maybe an occasional click from the relay. I replaced the battery, and then checked the charging action of it. Turned out that the new battery wasn't being charged properly either - replacing the voltage regulator brought the charging voltage up to 14V where is should be.
So, it is good that charging the battery and checking the cabling connections seems to have fixed your problem. But, also check the voltage at your battery terminals with the engine running ~4000 RPM and make sure you are getting sufficient charging voltage at the terminals (13.8V - 14.3V), no more, no less.
The tach swinging upwards I have found to be a helpful troubleshooting indicator - since the tach pickup is on the low (primary) side of the coil, it indicates that there is at least some voltage there, and the ICU is trying to operate.... what happens in the tach swinging upwards case is the ICU is struggling to bring the voltage up to proper 12V DC level, but as soon as the current draw gets above some small amount the voltage drops, which is picked up by the tach as a fire pulse. The current draw momentarily drops, allowing the voltage to rise up again, the ICU tries to bring the voltage back up to 12V, and the cycle repeats. Depending on the battery condition, etc. this can happen for hundreds of times over a few seconds, and make the tach swing upward. Eventually the voltage drops low enough that the ICU cannot bring the coil primary up high enough to register on the tach, and/or there isn't enough system voltage to run the tach and the idiot lights.