Sorry to add any confusion. My goal is to help you determine if the AirPort Express Base Station (AX), indeed, is the culprit.
I suggest that you start by looking at the Internet Connection tab within the AirPort Utility to identify your current PPPoE settings. (ref: AirPort Utility (v5.x) > Select the AX > Manual Setup > Internet > Internet Connection)
It should look something like the following. (Note: I don't use PPPoE and this screen shot is from the Windows version of the utility, but yours should look similar.)
I highlighted what would be your PPPoE User Credentials. Currently your AX should be configured to provide these in order for you to gain Internet access from your ISP. If you connect your Mac directly to the modem, your Mac will now need to provide these same credentials. The idea is to copy these done so that they are handy, and then, temporarily reconfigure your Mac to provide these. You would configure these setting within System Preferences > Network > Wi-Fi or AirPort