Just because a lack of immigrants were not the cause of the crisis, does not mean they couldn't have been a solution, much like how medicine does not cause a disease, but it sure as hell can cure it. In that case, certainly one would classify one's inability to cure the disease as a drawback. Thus, the entire answer by that poster can be disregarded unless they can adequately posit that immigrants would not have made any difference during that period in Japan.
Further, the original question is not solely focused on the crisis, but asks in general what drawbacks there have been in Japan due to a lack of immigration.
Speaking generally, it is well known that Japan is an ageing society, by which I mean that in Japan, many people live longer. With 74.9 years, it has the highest life expectancy in the world. In 1950, 4.9 % of the Japanese population were 65 years or older. In 2015, that number was 26.7 %. In 2050, that number will be 38.8 %. This naturally poses a problem which many western countries are facing as well: the dependency burden. The burden is just a much bigger problem for Japan. For example in the United States, only 15 % of the population in 2015 was 65 years or older, substantially less than the Japanese number. For that reason, this is not a "future" problem for Japan either, it's already a problem and has been for a while.
So, how do you solve that problem? You ensure that your labor force is both strong enough and educated enough so adequate production can be reached. How do you do that? Well, have more babies, educate more people ... and immigrants!. This is where the immigration policy has been a drawback for Japan, and they realize it as well: immigration in Japan has skyrocketed over past few decades, so that they can increase the labor force.
It is also likely that this dependency burden also played its role during the stagnation since the 2000s. The GDP per capita stagnated as we all know, but something weird was happening with the unemployment rate. You'd expect it to be rising during a crisis, yes? It actually fell:
This suggests that jobs were largely available in Japan, and the immigrants that were being brought in (as indicated by the purple chart above) were being put to good use, which ties in nicely with the demographic analysis above. There are simply too many old people in Japan, so immigrants can come in, get a job easily, and take some of the burden and contribute quickly to the production. Had more immigrants been allowed in, both in the past, during this period, and in the future, obviously Japan would have been better prepared to face some of these issues.
This entire answer should of course not come as a surprise: the primary thing immigrants offer are labor supply, and a lack of immigration has thus one primary drawback: lack of labor. Duh.
You claim in the comments that this added labor would not solve Japan's problems permanently, since immigrants would adopt the birth rates of ethnic Japanese. First of all, Japan is not an ageing society due to their birth rate (which is actually on the rise either way, from 1.26 in 2005 to 1.5 in 2016), but due to their high life expectancy which itself is a byproduct of a culture that generally respects the elderly far more than western countries do. Secondly, almost all solutions in politics are temporary solutions. Thirdly, even if the immigrants brought in today soon whither away, they can just bring in more immigrants in the future.