Italian bikes are not renowned for their reliability, especially on the elctronics. Troll the Ducati forums and see how many people have had their ECU take a dump for no reason and leave them with a $2000 bill to replace the ECU/gauge cluster mated pair because the immobilizer won't let the bike start. Fortunately, a few intrepid souls have hacked this system and can now defeat the immobilizer. Good news for owners who get screwed by flakey electronics or bought a junk bike with no original key, bad news for everyone since thieves now have the same unfettered access. I digress...
I wonder if the tech was making something up to justify the bill or sincerely passing on misinformation he got from someone else. All a push start does is rotate the engine just like the starter would do. If I get a Guzzi, Ducati, or Vespa and it needs a push start I'm going to push it.
I wouldn't worry too much on the Enfields, we don't have Italian electronics and I don't think the Indian designers who so relish the reliability reputation of the Bullet would design in such a fragile system.
Scott