I too think that it's not (only) about the popularity of the acts, but it's about power and connections as well.
I mean, in 2013/14, MM were close to Do it now/Koko ni Iruzee concerning first week sales, and on a level with Shabondama concerning total sales. I'd consider the time when 5ki/6ki entered still a very popular time for them TV-wise. And even though they've reached those sales again, they got nothing special out of it from television. Even with open declarations like "Our goal is to go to Kohaku this year!", Kohaku ignored them. So it's definitely not about sales.
IIRC, the whole thing about them not being invited is that they did have a huge argument for being invited up until the release of the The Best! ~Updated Morning Musume~ album. It sold a little over 16,000 copies and came in at #6 first week in comparison to Wagamama selling 140,000+ copies first week even though the best album was released only a month later, which called into question their popularity and the ~legitimacy of idol sales~. I still think releasing that album killed their chances of going on that year.
Besides Kouhaku while still a prestigious show it's not as big of a deal as it used to be. Shunkan Bunshun, a popular tabloid, did a poll last month and asked their readers who they wanted to see on Kouhaku and who they didn't want to see. A majority of acts people DO want to see are either retiring/retired/declined the Kouhaku invitation and a majority of the acts people don't want to see are...actually on the list lol.