Even traditional opposition to contractions isn't as broad and monolithic as most people imagine. For example, I've never heard that modern formal writing opposes the use of good-bye (from "God be with ye") or Halloween (from "All Hallow Even") or o'clock (from "of the clock"). More recently, I suspect, many people writing for scholarly or professional audiences allow themselves to use e-mail (from "electronic mail") and T-bill (from "Treasury bill"), among other contracted forms.
Usage commentators who condemn contractions tend to focus on two classes of such terms: contractions that include an apostrophe to signify the omission of one or more letters (such as I'm, don't, and we'll) and contractions that appear closed up but (often) duplicate a late consonant and (almost always) end in a vowel (such as coulda, gimme, gonna, gotta, hafta, oughtta, outta, sorta, and wanna).
I've noticed a tendency to blame opposition to free use of contractions on self-appointed grammar and style experts. But I think at least some of the blame rests with dictionaries, which have been very slow to acknowledge the existence of many such terms. In particular, Merriam-Webster's didn't begin to acknowledge many common apostrophe-containing contractions (including don't, I'm, o'clock, and we'll) until the Third Collegiate Dictionary (1916). And—as I discuss in my answer to How often do people say "gotta", "wanna" or "gonna" in English speaking countries?, the Eleventh Collegiate (2003) doesn't provide entries for such common terms as gimme, gonna, gotta, and wanna as contractions for "give me," "going to," "got to," and "want to," respectively.
These omissions must surely reflect some level of disapproval by the lexicographers at Merriam-Webster's with regard to the validity of the common words thus ignored, although I have no doubt that a desire to avoid opening the floodgates to a whole class of previously omitted words that would then have to be covered in the dictionary plays a role as well.
It is hardly surprising that words that fell into this underclass until less than a century ago—or that fall into the same underclass today—are objects of hostility from authorities on polite, formal, are scholarly writing. The process of reinforcement is ultimately circular, however, which means that don't, I'm, and we'll, are well on their way to acceptance in formal writing after 97 years of rehabilitation (and eventual approval) by Merriam-Webster's and other dictionaries, but also that gimme, gotta, and wanna have a long way to go.