[Sennheiser GSA-70] arecord lists USB sound card but cannot record: audio open error: No such file or directory

I suggest you consider updating BIOS to ‘Improve system performance’
ROG Crosshair VIII Hero (WI-FI) | rog.asus.com

The arecord error is correct that there is no device hw:0,0 because HDMI device is shown as card 0 and does not have a capture subdevice
(a USB device would normally be blocked from taking card 0 status by ALSA to allow for HDMI and onboard audio)

The USB Device is now shown as Card 2, but the card number could change if the 3 USB devices are detected in a different order
I suggest you use the card alias name instead of the card number --device="hw:E70,0"

I also suggest add options -c 1 for the mono channel map and -t wav

arecord -d 5 -c 1 -f S16_LE -r 16000 --device="hw:E70,0" -t wav rec.wav

I had hoped that this device would support a better sample rate than 16000Hz but it doesn’t

Sennheiser EPOS GSA 70 at usb-0000:08:00.3-4, full speed : USB Audio
Capture:
  Status: Stop
  Interface 4
    Altset 1
    Format: S16_LE
    Channels: 1
    Endpoint: 3 IN (NONE)
    Rates: 16000
    Bits: 16
    Channel map: MONO
Sennheiser EPOS GSA 70 at usb-0000:08:00.3-4, full speed : USB Audio #1