Bad idea to go cutting off those connectors unless you really know what you're doing. You leave stray wire ends hanging round just waiting to pick up water and short out, and you remove any possibility of you (or a future owner) changing your mind and installing an alarm, driving lights, charge socket, etc. It's not as if having unused connectors "lets electricity leak out" etc either - there's no harm in leaving them there. If you don't like the look, just tuck them away somewhere they can't be seen.
The reason cutting your alarm connector off stops your bike working is because both the starter motor and ignition circuits go though that connector so the alarm can disable them as part of its security. If you check out the blanking plug that was plugged into the connector it had two wire loops on it to keep these circuits operational. To fix it you need to locate the two pairs of important wires and connect them up - this will be made especially difficult by the fact that ALL the wires on the alarm connector are black. This is to stop thieves doing exactly what you're trying to do.
I -did- cut off my alarm connector, but I'm one of the ones who fall into the "really know what you're doing" category. I removed mine because I was fitting an aftermarket alarm and at the time I had no source for the connectors so I hard wired the alarm in. I've since found a source for the connectors and have both a new plug and socket, but I see no real reason to mess with it while it's working. If the alarm dies I'll put the new socket on so I can fit a standard alarm.
