In layman's terms, Emperor Meiji was an individual educated person being used as a symbol. So in some sense he was a "puppet" of social currents way beyond his control, and as far as I am aware, he never overruled his cabinet as an absolute monarch would. By acknowledging the symbolic power of his hereditary position, he reduced the political responsibility of the imperial throne to the extent that his son Emperor Taishō did not need to contribute anything to the state at all. However, he also did a lot of other stuff and was extremely busy, and the term "symbol" needs some qualifications.

During the Meiji Restoration and Boshin War, the pro-emperor side was rallying to the symbolism of the imperial household. Their politics was not grounded in the personality of any individual emperor, but befriending the actual emperor was a kind of necessary contingency for their movement. Luckily, Emperor Meiji was a fairly intelligent person and understood the changing needs of Japanese society. He was not a rigid opponent to Westernization like his father Emperor Kōmei (although I must caution that we have limited information about Kōmei's private thoughts, and at the time of his reign, the pro-emperor faction would have been highly anti-foreign).

What was Meiji's job description in the post-1868 government? He was not actually in charge of the myriad things that needed to be done politically, like building the legitimacy and stability of the new regime, modernizing the military, establishing universal education, etc. In the early years, Meiji was at saddled with a bewildering and kind of ridiculous number of ritual duties, both continuing on actual ancient traditions of the imperial household and other ceremonial duties that ideological extremists of the new regime wanted him to conduct personally, including rituals previously performed by subordinates, resurrected "traditions" they had found in old books, and even completely new rituals they were literally making up. This would have occupied much of his time during the turbulent first years of the Meiji era. As the ideological extremists faded in power, these Shinto-like ritual duties were replaced with a new, more relevant set of duties for international politics, learning a completely new and foreign set of etiquette protocols incumbent on a Western head of state.

Despite this busy schedule Meiji had time to keep abreast of Japan's rapidly changing domestic and foreign political issues and frequently offered comments to his "advisors", i.e. the cabinet members who were actually doing his administrative work. I can't think of any brilliant insights he had off the top of my head, but the details are documented in Donald Keene's massive book Emperor of Japan: Meiji and His World, 1852-1912. Meiji was also constantly meeting with domestic and foreign dignitaries, writing poetry, and overseeing non-ritual household affairs such as the remodeling of his new palace in Tokyo and the education of his children.

Basically, even though Meiji was not the architect of the Meiji regime, he was a very busy person. His son Taishō, who is rumored to have been intellectually disabled (the truth of this is unknown), reigned over a period of relative stability and was not so busy. So what was the nature of the imperial institution from Meiji to 1945? I characterized Meiji as a "symbol" but at the time, some Japanese people were very angry with the concept of emperor-as-symbol. To a certain political faction in imperial Japan, the emperor was meant to be an absolute monarch, meaning that Meiji and Taishō were not displaying their full power level, so to speak. In practical terms this was never really possible and even Hirohito, who reigned over the wartime period when the emperor was seen as most important, had limited ability to overrule his advisors.