Up until September of last year, the Prime Minister was Shinzo Abe. PM Abe was a total asshole in many ways (he just couldn’t stop visiting Yasukuni Shrine…you know, the one with all the war criminals interred there), but his government at least gave the sense that it was doing something, anything.
After Abe stepped down (officially due to health reasons but also with a lot of corruption scandals surrounding his administration), his hand-picked, right-hand man Yoshihide Suga was elected by the ruling party, the Liberal Democratic Party (which is none of those things). Suga was a great right-hand man, but he is clearly out of his element as the leader of a country, and has barely done anything at all in his 8 months to date as PM.
Japan has not done any lockdowns; instead they have declared three or four “States of Emergency,” under which non-essential businesses were asked kindly to shut down and offered cash incentives to do so. They only recently added fines (this February), but compliance has nevertheless been pretty high (bars and restaurants remaining open and ignoring the guidelines tend to make the news). The problem is that they keep lifting the state of emergency too quickly, so infections surge and they have to start all over again. (Incidentally, there has never been a mask mandate, but mask compliance is virtually 100%. I almost never see anyone not wearing a mask.) The framework for all of this is a leftover from the Abe administration; PM Suga has not come up with anything at all.
As for the slow pace of the vaccine rollout, aside from poor logistics (they secured commitments for doses, but they never thought about how to actually get those doses from the manufacturers to Japan), one major issue is that they are insisting on doing clinical trials here in Japan rather than just using the data from trials conducted overseas. Now that they have more vaccine doses on hand, they are leaving it entirely up to municipalities to arrange the actual vaccination, but municipalities are totally unprepared for this. Towns and cities have to identify residents who qualify (by age, etc.), contact those residents and then schedule the actual vaccination. It’s a total mess…hospitals have to put up signs saying, “We do not know when vaccines will be available. Please wait to receive a letter from the city.” The central government had plenty of time to plan for all of this while they were working on approving the vaccine to begin with, but like I said at the beginning, PM Suga has done absolutely nothing while in office. The governor of Tokyo (Yuriko Koike, whom a lot of people think should be the next PM) has done more for the entire country than the goddamn prime minister has.
ETA: I am in Kobe, which is about 30 minutes west of Osaka by train. Osaka and Kobe have been hit hard by mutant variants of the virus (the UK variant and now the Indian variant), so the situation has been worse here over the past month or so than in Tokyo. Nevertheless, the fact that only 5~6,000 new cases are appearing daily nationwide is a testament to how effective masks and social distancing can be.