Early true BIOSen were stored on EPROM, which couldn’t be rewritten while on the board, so those were read-only.
Later BIOSen were often on EEPROM or other chips that could be reflashed while on the board. According to Wikipedia, that started in the mid-1990s. However, you usually needed physical access and/or special software tools to do an overwrite—you couldn’t mount these as a filesystem.
UEFI is quite different from legacy BIOSen and can be mounted as a filesystem, but how much it can be tampered with varies between implementations and devices.
So you would have been correct up until about 30 years ago, but not for modern systems.