You can actually do a New Game + without a script, but it's massively tedious to program. Teleport the character back to the start of the game upon completion, reset all your switches, reset all your variables, reset anything else you don't want them to have, then the game starts over.
It is seriously massively tedious to program though (I did it once in a 15 minute game just to see if it could be done... took a grand total of 8 hours to hammer out everything I wanted to reset).
To be honest, I've always wondered why people used the "New Game Plus" feature to try to extend their game. No, I don't mean, extend the playtime value. I mean, using it as a freakin' brick wall to block players off from content until they've beaten the game once. I've always thought this was kind of silly. Why not include content in the game the serves the same purpose without the New Game Plus mode if that's what you want? It's not hard to simply implement harder monsters and harder areas in the game for those close to the end. It's also not hard to create unlockable areas within the game that only open up once you reach certain stats or levels. Why lock off any new content behind the New Game Plus wall at all? It doesn't need to be behind that wall.
The point of a New Game + is to play through the game again, making different decisions, or collecting things you missed while using your levels and stats to easily breeze through most of the content. In Chrono Trigger, it's used for exactly that, to make choices you didn't make before. It's there to let you obtain endings you didn't on the first playthrough or the second or third playthrough. It's also there to let you make different choices on which loot to pick up at certain instances. Oh, and the other reason it's there? Help you max out your characters and obtain more of the most powerful (but really limited) equipment.
In Mass Effect, that is the same point of the "New Game Plus". It's there to let you make new decisions if you want, continue exploring, continue getting more powerful. The only content locked off behind the "New Game Plus" wall in the first Mass Effect is the ability to get to level 60 instead of being level 50.
If you're locking off entire sections of your game and require a second playthrough to even access them, your game is going to be somewhat frustrating to players. Hell, I get frustrated when the hardest difficulty is locked behind the wall of "beat the game once". Why is this content not just available from the start? On top of which, locking off this content doesn't guarantee players will go find it or complete it. If your game was boring and players hated the first playthrough, they're not going to play through it again just to experience all the locked off content. It's a gamble when you do things like that.
Basically, only include New Game Plus if you want a kind of "cheat mode" for the game after you've already beaten it, or if you want players to explore different things than they explored the first time. Otherwise... it's kind of silly to have it just to lock content behind it.