Firefox

A "backdoor" is just an undocumented API. It may have security issues, but it's still an API.

And yeah, companies love their backdoors because it makes a lot of things easier for the vendor. Apple loves the ability to send security patches without the user being aware so they can silently patch systems to limit the impact of a virus. That's still a backdoor, but its arguably a desirable one. Microsoft used its control of the OS to make other browsers run worse than IE, and that resulted in the antitrust case of 1998, after which Microsoft agreed to share its APIs. That's arguably an undesirable use of a backdoor.

So why do they exist? Mostly to make things more convenient for the vendor. Sometimes it's in the users interest, sometimes it's not, but it's never made to support external criminal enterprise and rarely made to support internal criminal enterprise. Usually it's just to cut costs.