The trustworthy exchanges don't have this system for listing a coin for just 1-2 weeks. If you want to list your project in a major exchange such as Binance or Huobi, then you need to pay the listing fee (which may amount to several million USD). On top of that, there will be very careful screening of the project from these exchanges and if the inspectors flag the project as shady, then they will return the listing fee and refuse to deal with the project. So in 99% of the cases, if a project gets listed in Binance or Huobi, we can be assured that it is a genuine project.
They all used to actually,,, but you are right now there do last for a few months but I have personally seen a big exchange like Bittrex only list for a few weeks, especially with their IEO tokens that only listed for a few days before price crashed to even -99% of the IEO price, then they just removed it because of all the FUD.