A brand new $70 carb is not going to work better than a clean rebuilt carb. Or, rarely will it.
To the OP: there could be lots and lots of reasons it won't idle. If the carb is clean and has no leaks, it could be an air leak at the carb connection, or at the crank seal. It could also be jetted incorrectly if the previous owner "hot rodded" it.
That phrase is really unhelpful, as I'm sure you know. There are literally dozens of things that can be done to improve these bikes. Any one of which could be called "hotrodding". There are also dozens of completely stupid mods that people do to mopeds believing that they're magically going to create 100hp and 70mph. And those mods often make the bike run worse, if they have any effect at all.
Your best bet is to figure out what stock was, try to match that, and then go from there.
Unfortunately, you're going to have to read bunches of stuff in the wiki, or Myrons Moped website, ask bunches of questions, get bunches of answers to sift, and then just end up having to go by trial and error until you figure it out.
Fortunately for you, all of that is the good stuff about mopeds. It'll be a learning process that makes you feel good when you succeed (and like crap when you fail). It'll be how you make moped friends in the community. It'll be how you get addicted and end up with 10 in your garage and a $4000 wish list on treatland.
Welcome to mopeds. (edited)