I don't see any evidence to support this assertion.
Non-robot anime was produced with limited animation for years before the debut of Tetsujin 28 in 1963 or (the notably limited) Mazinger Z in 1972. Even today, lower-budget anime use pan cels or "talking head" scenes to communicate story via human characters while economizing on movement.
As for the actual examples raised in that quote, they don't hold up very well.
Space Battleship Yamato's writer-director-designer, Leiji Matsumoto, has always emphasized the artistry of animation over the constraints of working within limits. For instance, in 1978 he told Kinejun magazine,
...no matter what, the animation of Disney and Fleischer still hold an
immovable influence in my mind. If one wants to make exceptional
animation, you look at Disney in particular. Bambi and Pinocchio are
the result of human blood and sweat. You can feel the youth, passion,
and vitality of Disney in them.
(translation by Tim Eldred)
Yamato is one example among many of Matsumoto's aesthetic preferences. Matsumoto was the driving force behind the project, and like most of his creations, it's a space opera featuring a terrestrial vehicle transposed to space. Galaxy Express 999 has its train, Captain Harlock has his own many-turreted space battleship, Queen Emeraldas has her space dirigible, and the Yamato was raised off the ocean floor to become a space battleship.
There is no evidence that Matsumoto chose spaceships as a theme of his work, throughout his career, in order to save on animation costs.
Mobile Suit Gundam is a curious example because (like many anime productions) it had no singular impetus, but a combination of factors inside and outside Nippon Sunrise which influenced its production. Gundam director Yoshiyuki Tomino was coming off a successful run at Sunrise on the giant robot series Zambot 3 and Daitarn 3 in 1977 and '78, and he had the desire to create something more serious and adult. He was inspired, in particular, by Heinlein's Starship Troopers, and its concept of "mobile infantry" (this is pretty well documented in many Japanese books, such as Gundam Century, cited here by translator Mark Simmons). He originally intended for Gundam to focus on powered armor, not giant robots.
From what we can see in the early design art for Gundam (as published in the Mobile Suit Gundam Roman Album, among others), the "Gunboy" was at first a more streamlined machine with a smooth faceplate like that of a Mercury-era space suit. Later, when the series' development shifted toward giant robots, the design shifted toward the one we know today, with its two eyes and samurai-like forehead crest (reminiscent of Zambot and Daitarn). The reason for this change was simple: the sponsorship of the toy company Clover. Clover's toys emphasized the combining aspects of the Gundam and its optional equipment, as well as additional weapons not featured in the anime, such as shoulder-mounted missile launchers and lances. The effects of toy sponsorship on the anime were such that, while the heroes' mobile suits were still treated as weapons of war, they were now unique and heroic in stature.
In short, the conception and design of giant robots in Gundam was not driven by animation concerns - it was a tug-of-war between the director's desire to use grounded science fiction concepts and the priorities of the series' toy company sponsor.
It's certainly possible that some anime studios at some point chose to create mecha anime because of the ease of animation, the potential to reuse stock footage each week, etc. But regarding the examples cited by the TVTropes article - Yamato and Gundam - there's no information to justify that claim. Like most mecha anime, they were motivated by creators' personal aesthetics, creators' love of science fiction, and sponsors' desire to sell toys.