RE: It shouldn't be a problem with the host either, because it does work using other contact plugins
Sorry but that is not a valid conclusion. There are a host of issues that can occur with sending mail. They depend partly on how a plugin sends mail but far more importantly your particular WordPress installation.
Though the plugin successfully sends millions of emails every day, there are a host of issues that can delay or stop emails on both the sending and receiving ends of a typical email. It depends entirely on your local configuration.
You will need to investigate this issue for your particular local configuration.