Gorim said:A great question! Lets see if I can offer some insight to this. I majored in Japanese history back in the day, but post-war era (and its film history) are certainly not my field of specialty.
First thing to realize is that anime is far different from Japanese film and television. I think Japanese film industry and television certainly depicted and continue to depict these decades. When I think of post-war Showa-era films and television in Japan there are of course Kurosawa with his samurai movies, but more than that I remember Yasujiro Ozu with his depiction of real people and their troubles, Kenji Mizoguchi with his societal critique or Otoko wa Tsurai yo -type of everyday life and its misfortunes films. Anime except something like Shouwa monogatari, Sazae-san or Tonari no Yamada-kun doesn't really follow the same themes.
Economical realities in making historical series
To get back to the question. As you say, not many historical dramas in anime take place in 60's and 70's. This is mainly due to the fact that there aren't too many historical anime in the first place. Anime is difficult and expensive to make which is why most of the time only the most popular manga/ranobe/etc. get an anime adaptation. As a historian I am sad to say, but historical manga is rarely ever popular enough to get an adaptation. This is why a lot of historical manga never get an anime adaptation and the ones that get published often are tales pre-Showa Japan.
"Why not make a historical series of the post-war era then?", you ask. Another great question. The simple answer of course is that there are series made about the subject (Rainbow being one of the best historical anime ever made). The reason why there are fewer compared to Sengoku-, Edo- or Meiji-era is simply because 1950's&1960's aren't as hot topics (from historical point of view) as the older time periods are. Many people want to watch historical dramas because history holds a sort of 'fantasy series' type of feeling for the viewer. The series tell a tale about era which the viewer never experienced a tale about a historical landscape far different from our own. The closer the time period is to our own the less fantastical it feels. Something like Rainbow certainly feels shocking for our contemporary Japanese, because it differs so much from everyday life that they live (just like "Soredemo boku wa yattenai" shocked Japan in 2007).
There of course might be multiple other reasons such as: if the time period is too close to or own many of the viewers might be turned off because the subject matter hits too close to home. This can be a good or a bad thing, but because people who fund anime (mainly manga/ranobe companies such as Kadokawa/Kodansha) try to maximum the profits and reduce the risks as low as possible they won't give funds for projects that won't reach as wide audience as possible.
Things like samurais, Meiji/Taisho-era and their fashion, WW2 and political turmoil during Bakumatsu have already their established viewers so its easier to anticipate certain level of viewers by simply offering a new series about the same era. Same can not be said about post-war Shouwa.
History of 1950's-1970's impacts many different anime
Yes, like you mentioned there are a few series already which are set in 60's&70's. I'd like to name a few more: Shouwa Genroku Rakugo Shinjuu, Ushiro no Shoumen Daare, Shouwa Monogatari as well as few others. However even if we listed them all, the list would still be numerically lacking and the series in there would be too obscure to make a real difference.
However! The era from 1950's to 1970's hold an important significance in Japanese history as a political era. In the west we have a far different point-of-view to that era than the Japanese have. For us the post-war Japan represents an economical miracle and growth to prosperity. For many Japanese it represents a bitter history. Economical growth at the cost of environment, growing income inequality in the name of individualism, harsh poverty for millions, transformation from imperial era to corrupt de facto one party politics, U.S. supremacy over nations own foreign policies & military matters. Etc.
1950's and 1960's in Japanese history are defined by its activist movements. Environment movement, anti-nuclear movement, socialist activism in 50's, anti-war movement and of course university student activism. Assassination of prime minister Inejiro Asanuma and the public seppuku by author Yukio Mishima were era defining moments.
Especially the anti-war, anti-nuclear and environment movements have inspired multiple authors to make anime about them. Hayao Miyazaki being one of them. For example Tonari no Totoro takes place in 1958 in a rural beautiful landscape, but like in most Miyazaki-films there is an underlying critique towards modernization. That beautiful landscape was bulldozed and made in to a concrete jungle in the 1970's and now only Miyazakis film reminds of us what used to be there before the concrete apartment buildings.
The subtle (and often not so subtle) environment critique can be found everywhere in anime. If we think about anime made during the time period Uchuu Senkan Yamato certainly is one of the most prominent representatives of the national feelings towards anti-nuclear & pro-environment agenda. The series pawed way for other classics such as anti-war themed original Mobile Suit Gundam by Yoshiyuki Tomino. And of course outside anime things like Godzilla and other kaijuu-films warned us about the dangers of pollution and nuclear power.
Modern anime creators have watched these fictions and even though the anti-war/pro-environment Sci-Fi & fantasy series's are not set in 60's or 70's they often straight up parallel the real historical events through the lens of fiction . Chikyuu Shoujo Arjuna's creator Shouji Kawamori is one of the directors who grew up during environment activist era and has picked up a lot of things from his predecessors.