[gruez]

Do mirrorless cameras even have physical shutters? Or is it all digital?

[sudosysgen]

Older ones do, because the readout speed of sensors was not fast enough. Cameras with stacked or global shutters generally don't, because it's no longer an advantage. In between, it's a choice you can make whether or not to use the shutter.

[yakubin]

Most do. Some, like Nikon Z8 and Z9, don't.