https://www.sport-express.ru/olympics/sochi2014/figure-skating/reviews/823131/
Here's an article called "Multiway substitutions". It was published on February 8, 2014 - the morning of ladies SP in the team event of Sochi Olympics. "Sport Express" is a national newspaper, a respectable one at that..., not a tabloid or something. The author of this article is Elena Vaytsekhovskaya - one of THE most respected Russian sports journalists. She's an Olympic champion herself (platform diving, Montreal, 1976).
So, first, I translate the part of the article which directly talks about what led to the substitution of ONE ice dance pair to TWO pairs, which consequently led to one less ladies entry:
"When I asked whether that was the final decision of Russian Figure Skating Federation (RFSF) to have only one ladies entry in the team event, the president of RFSF Aleksandr Gorshkov answered evasively: you know, Olympic Games - they are unpredictable. Anyone can get sick or get injured, that's why the deadline for submitting the official entries is set as close to the performance as possible. Americans, for example, based on what we saw during the training sessions, were going to submit the American champion Gracie Gold for the SP, but eventually it turned out that Ashley Wagner would skate in this section of the competition.
A few days before the beginning of the Games I heard another story from one of the coaches: that during the test skate, which was held for Evgeni Pluschenko in Novogorsk after European championships, experts were left somewhat disappointed with the performance of Ekaterina Bobrova and Dmitri Soloviev. And that supposedly this was the moment when a behind-the-scenes decision was made to substitute the skaters of Aleksander Zhulin [Bobrova/Soloviev] in the team ice dance event, giving them the short program only.
It was also talked about the second Russian ice dance team, Elena Ilinykh and Nikita Katsalapov, that there was an influential sponsorship company who wanted to see this pair on the ice very much. Elena was the "face" of this company during the Olympic season, and it influenced the decision of the federation.
This version of events, you see, is perfectly viable. Moreover, because it's obvious: by substituting one ice dance team for the other - RFSF doesn't risk anything because the strength of both dance teams is rather equal."
Hopefully this clears up some doubt that actually there WAS certain "politicking" going on...
In this particular article, there's no direct reference to the Minister of Sports, Vitaly Mutko, what role he played in these decisions and his phone call with Tutberidze. In case I find an explicit reference somewhere else - I can share that, especially for you..., because I remember PERFECTLY WELL that Vaytsekhovskaya talked about it. Furthermore, this greatly contributed to Vaytsekhovskaya's long-standing feud with Tutberidze, which is a whole another story.
(Vaytsekhovskaya jumped to Lipnitskaya's defense, criticizing Tutberidze for burning out the 15-y.o. by contributing to the decision to make her skate both programs; about not taking care of Lipnitskaya's eating disorder which was an open secret at the time, etc. etc. Let's just say... Tutberidze was NOT happy.)
During and after the Games, Vaytsekhovskaya also shared a lot of behind-the-scenes info on her blog. Unfortunately those comments are not searchable, and there are thousands upon thousands of comments... I don't 100% remember regarding every single piece of information - whether I read that in some newspaper, organization web page or a blog.
The other significant part of this article is regarding Lipnitskaya - was she UNANIMOUSLY the favorite for the only ladies entry in the team event, as potentially the most consistent skater. On the contrary of what some fans have argued here during the last few days, among Russian figure skating experts - in fact, she was not. At least, not unanimously. As you can read in the translation above, even the president of RFSF just ONE DAY before the event didn't give a clear "yes, Lipnitskaya is the favorite for the team event, that's her who will skate".
"By the way, by far not everyone whom I talked to was convinced that among ladies singles skaters Lipnitskaya would be the one who had a better chance at surviving the Olympic marathon. One of them, for example, had exactly the opposite opinion - Adelina has been skating at senior level for 6 years already, she has become the national champion 4 times, 2 times European silver medalist, therefore the figure skating world perceives her as a mature, accomplished athlete. Meanwhile, there are not so many successful competitions on Lipnitskaya's resume."