How to fix when an Android phone stopped showing the “mass storage” connection option?

Just rebooting the phone should fix this if this option is at least usually shown when connecting the phone by USB cable to a computer.

It does not help to use the feature in settings to "Safely Remove SD Card" and then drawing the SD card out and plugging it in again – the SD card is accessible by the phone without problems still, just not shared to a computer.

It might be that switching the "USB Tethering" feature on and off multiple times is what triggers this issue (since, by design, the USB mass storage connection option is not shown while USB Tethering is enabled on the phone; and by error, maybe also not afterwards in some cases).