Is there a reason for the prefix change of in-/un- in about the 60s period for these words?

In broad strokes, Doug Harper (etymonline.com) sketches the historical contest between the negating prefixes un- and in-:

un- The most prolific of English prefixes, freely and widely used in Old English, where it forms more than 1,000 compounds. It underwent a mass extinction in early Middle English, but emerged with renewed vigor 16c. to form compounds with native and imported words. It disputes with Latin-derived cognate in- the right to form the negation of certain words (indigestable/undigestable, etc.), and though both might be deployed in cooperation to indicate shades of meaning (unfamous/infamous), typically they are not.

A deciding factor in this battle is often who gets there first. An adjective from French or Latin wouldn’t always bring along its negative twin, or the noun arrives with no adjective attached. A native un- might be prefixed, then elite speakers might prefer something that looked like their Latin lessons. Sometimes they won; sometimes they lost; sometimes they didn’t bother. As Harper notes, occasionally there was even a truce and both forms coexisted. This means, unfortunately, that each word and its negatives would have to be individually traced to the present day.

Another factor is a semantic one: a word like inequality, imbalance or injustice defines a quality or state; unequal, unbalanced and unjust describe an absence. If that absence is of something good or desirable, then there may be a tendency to choose un- over an original in-. An example with another prefix: assymetry is a characteristic of much modern architecture and design, but if something is unsymmetrical, I might ask you to move that portrait of your grandmother a bit to the right.

The third factor, and really the only one, is which people use which word and how many people imitate them. One variant, for example, may have met its demise simply because Lindley Murray didn’t like it.

Popular, Unpopular, Impopular

Popular hopped across the Channel in the early 15th c. from MidFr populier, but only in the sense ‘of the people’, a quality/state negated with not:

megabisus therefore disliked a democraticall comon wealth, … his reason tended to haue a common wealth called oligarchia, or aristocratia, and not a popular state: — Ludowick Lloyd, The consent of time disciphering the errors of the Grecians in their Olympiads, 1590. EEBO

A new sense, however, was on the horizon:

by labouring to be popular, he becometh the ringleader of all mutinies in the house,… — Christopher Bagshaw, A true relation of the faction begun at VVisbich, 1595, 1601. EEBO

which could very easily be negated with un-:

…witnesses to partial men, that in this work I had not given the worst experiment of an industry joyn'd with integrity and the free utterance though of an unpopular truth. — John Milton, Tetrachordon, 1645.

But in Latin, and meanwhile French, there was a negative waiting to be “Englished”:

I knew well the Diſadvantage upon which I ſpoke in ſo tender a Point, and too impopular a Thing it was to be againſt the convening of the Par­liament in a Time of ſo great Straits, … — Edward Hyde, 1st Earl of Clarendon, A Discourse by Way of Vindication, 24 July 1668.

But the cauſe being ſo very odious and impopular, the tryal of the Verdict was defer’d from one Term to another, … — Jonathan Swift, Letter to Alexander Pope, Dublin, 10 Jan. 1721.

This relatively new word, from Latin impopularis or French impopulair, so violated Lindley Murray’s sense of linguistic ”purity” — there was, after all, a perfectly useful English word that was good enough for Milton — that he offered this sentence for students to correct for style:

His natural severity rendered him a very impopular speaker. — Lindley Murray, English Exercises, London, 1802.

Murry is one of the most prolific — and plagiarized — grammarians of all time, and his impopular speaker reverberates across the entire nineteenth century, not only in revised editions of Murray’s own works, but in the anonymously published A Help to Young Writer, by a President of a College (Albany NY 1836); Peter Bullion’s Principles of English Grammar, (New York,1850) through various imprints and revisions; an Irish schoolbook An English grammar, for the use of schools (Dublin, 1863), all the way to Longmans School Composition (London, 1892) by Peter Salmon.

Not everyone got the memo:

… and the emperor has rendered himself impopular by abolishing the practice of wearing beards. — Indiana Gazette (Corydon, Harrison County), 30 Aug. 1821.

Today, of course, Murray can now rest easy; his impopular speaker is silent at last.

The Histories

Each word in your list will have a story like this one, though it won’t necessarily take the same journey. I had originally begun to trace how unviable becomes more frequent than inviable. The first station led to unviable seeds and spores in horticulture and the tragedy of unviable infants who die shortly after birth (end of 19th c.). This usage seemed appropriate because in the grand scheme of things, seeds should sprout and children thrive, i. e., that something unsymmetrical shouldn’t be.

From these two disciplines, unviable spread to other sciences and a few pioneer writers in the 1920s with a flair for metaphor. The un- variant finally was adopted by political science, economics, and diplomacy during WWII.

After the war unviable economic conditions in Germany and Austria were discussed with an eye toward the Soviet Union. Soon to be independent colonies had unviable economies and would need American foreign aid. And there were those who wondered about this new jargon. From there, however, it spread into general use as we know it today. Other fields, like genetics, still seem to favor inviable.

It wasn’t so much that writers who had previously used inviable suddenly switched, but that a lot more people were using it to describe a lot more unviable things with a word previously reserved to academics.

But this journey required more than three pages of single-spaced citations to map out, and offering them here even in an abbreviated form would likely exceed the character limit of the website and the patience of anyone who might wish to read it.