JavaScript is a strange beast. The semantics of the language are, in my opinion, awesome and awful in different ways. Awesome: prototypes, first-class functions, dynamic function scope (not sure if there's a proper name for this), jQuery. Awful: the this keyword is confusing (but powerful once you learn it), type coercion, the API. The standard API is the reason that people care about "the good parts" of JavaScript.…
document.getElementById() isn't part of JavaScript but part of the DOM Level 2 spec that came out years after JS.
JS was first seen in Netscape 2.0 in 1995 whereas DOM level 1 was specified in 1998 but didn't in fact yet contain document.getElementById. That one was added in DOM Level 2 in 2000, so five years after JS.
As such it was not designed by Brendan Eich, nor was it created in 10 days.
No. This is the work of a committee that was heavily influenced by the way Java does it which is understandable because when DOM1 came out it looked like Java is going to be the be all end all of Computer languages.
The "library" that comes with JS is really small (methods on String, RegEx, Date and Array). Everything else we owe to the W3C which has created an API that was all but made for the language it was going to be used from most of the time.
(edit: I got my dates and dom levels mixed up - fixed)