Can I access my Android through ADB if the phone can only get to the fastboot menu? I would like to recover message attachments from a Pixel 3 that died recently. (Yes, I should have had them saved and backed up, but I missed that option for message attachments.)

However, the phone is minimally functional. If the phone could turn on, I could sync those files and be done. It also appears that I need to turn the phone on to enable usb debugging for ADB. I can reach the fastboot menu by holding the power and volume down buttons, but if I try to go past that into "Start" or "Recovery Mode" or even "Rescue Mode," the phone powers off again.

While in the fastboot menu, I can see that a Pixel 3 device is connected by usb to my computer. I have installed Android Studio, Android SDK, and the Google USB drivers. The phone appeared as "LeMobile Android Device" at first, but after changing the driver to Google's it is now "Android Device"--"Android Bootloader Interface."

Running adb devices returns an empty list even while Device Manager shows it is attached (probably because I can't turn the phone on and enable usb debugging).

I'm out of ideas, and am looking for ADB tips to try. What can I try next? How can I get the phone to start or to get access to the phone without leaving the fastboot menu?

Thanks.
No you can't access your phone via adb while you are in fastboot menu.

While in fastboot menu only the fastboot binary + driver can access your device. 

ADB is only available when the device is booted (adb is usually already available from a certain boot stage on while the device is still booting). 

Also certain recovery ROMs provide full ADB access. Simple recovery ROMS only provide limited ADB access, for adb sideload - it allows you to upload ROM images so that the recovery can install it.