This seems to reflect what Data suggested, that there is an issue with the battery terminals not making a reliable connection.
I also seem to remember someone having terrible troubles with electricals, with the problem eventually being revealed as a damaged wire underneath the battery (or something along those lines).

Interesting the bike came to life when you gave the terminals a check. This can indicate that the new battery may have an issue with a loose battery pole inside the casing. Not always easy to detect. Or you may have a partially damage cable inside it's insulation. I've seen both of those plus many more issues centred around the battery.

Edit: forgot to say a proper battery full load test may reveal any internal issues with that battery.