Whatever this nuisance is, it appeared to begin when I got logitech gaming headphones. I don't play multiplayer much but did want to talk on Discord. Naturally I thought nothing of it being related to a game that had never been used with them.

But since then the game crashes everytime. Google revealed others with an issue with this happening if a mic isn't attached. But so far I can't find a solution that works for me. I have perfectly nice speakers I want to listen to and there is no one around to hear the game.

Does anyone know how I can make it work without the darn headphones?

You should be able to simply enable stereo mix to solve this issue. If you don't have stereo mix, make sure you check the option to show disabled devices. In the event that it still doesn't show up you will want to download stereo mix drivers for your operating system.

I had this very same problem but I found this information online, I hope it helps:

You may encounter audio problems or a game crash when attempting to play Star Wars Battlefront II on Windows XP SP3, VISTA, 7, 8/8.1 or 10 unless a microphone is plugged into the microphone jack of your computer. The same issue can occur if the "Stereo Mix" device is not enabled on the Recording Tab of the "Sound" Windows Control Panel applet. For additional information on these issues, and other troubleshooting tips, please refer to the Troubleshooting Guide document.

Why does my game crash when starting a new single player or joining a multiplayer game in Windows Vista® when audio is enabled?

Star Wars Battlefront II was developed prior to the release of the Windows Vista operating system. As a result, some users may experience an issue with starting new single player or joining multiplayer games in Windows Vista when the in-game audio is enabled. We have found that some users have been able to get around this issue by following one or more of the following procedures:

• By plugging in a Microphone into the Microphone jack on the computer. The Microphone jack is usually located on the back or the side of the computer as a small pink 3.5mm jack/input. It is generally close to where the speakers are plugged into the computer.
• Ensuring the latest audio drivers are installed on their system – especially if using Realtek or VIA HD chipsets. The default Windows Vista drivers are not sufficient in this case and must be upgraded (please refer to your motherboard manual and/or motherboard website for more information about upgrading your audio drivers in Windows Vista).
• Enabling the “Stereo Mix” device in the “Recording” tab of the Windows Control Panel “Sound” applet.
• Note: The “Stereo Mix” device will sometimes not be present depending on the audio drivers you have installed. Please be sure to check that “Show Disabled Devices” is checked in the Recording tab of your Windows Control Panel “Sound” applet by right clicking in the main area of the window of the “Recording” tab of the Sound applet and selecting “Show Disabled Devices”. Once the Disabled Devices are present in the list you should be able to see “Stereo Mix” (if available) and enable it.

GOG provided a fixed BattlefrontII.exe file (1.1 UK version) that fixes these problems (just one byte was different, actually).