This isn't quite an accurate summary of MMT, which isn't really relevant to the question either. Let's review a couple things.
Inflation occurs when the money supply grows more rapidly than the value of the goods and services in the real economy, shifting the balance of economic/financial power from net savers (generally people and corporations who were well-off to start with) to net borrowers--those who didn't have the money to pay cash up front to begin with--which in modern society includes the vast majority of young people (due to educational debt), aspiring homeowners, etc.
Controlled inflation (provided there is sufficient political power in the hands of workers and seniors on fixed incomes to demand cost of living increases) is usually healthy for an economy, because it disincentivizes hoarding. Money that will be worth less in the future is better off spent on stuff (demand for which drives increased output) or invested in productive capacity that generates a return (to meet that demand). Since you can't just sit on your money and watch it be worth more, you have to put the money to work; and since loans will be paid back with cheaper future-money, the risk of borrowing for a speculative enterprise is somewhat reduced. (Of course, lenders raise rates to compensate; there is no free lunch here. But on the whole, inflationary policy leads to looser and cheaper lending.)
On the other hand, without inflation, savers have no incentive to invest in increased productive capacity--lower inflation means less demand means less market for the goods produced, anyway! If your money will be worth more in the future just from sitting on it, you either sit on it, or invest in very stable investments with predictable returns (such as prime land ownership, which is unlikely to lose value, and which is especially attractive if there are fixed rents, since a deflationary environment means the monthly take will just be worth more and more). Deflationary environments also tend to lead to the concentration of existing productive resources (such as, again, prime land) in the hands of the wealthiest parts of society, since more marginal owners of productive resources are often forced by their increasing debt service obligations to sell at unfavorable terms.
Healthy inflation is distinct from runaway inflation or "hyperinflation"; but while this is much talked-about, it's extremely rare. All instances that I'm aware of--1930s Germany, modern Venezuela, Zimbabwe, etc--are the result of either a sustained shock to a country's real productive capacity (Zimbabwe), or the value of that output (Venezuela, Chile briefly), or the result of having excessive debts denominated in foreign currencies which cannot be acquired by the State due to balance-of-trade issues (Weimar Germany), or some combination of those. Hyperinflation is not the result of policy decisions.
Promoting inflation is not inherently related to MMT, nor is it an MMT policy goal per se. MMT is a description of how the economy actually operates. (Your linked Wikipedia article is good; you might see also here.
Specifically: governments create money by spending and destroy money through taxes (which also have the effect of creating demand for the money in the first place--everybody wants [dollars, euros, dinars...] because they know [the USA, the European governments, the various Ottoman successor states...] will be demanding them in taxes, which means a lot of other people will want to get their hands on them). To ensure price stability, the governments need to make sure this system creates money at the same rate as the economy creates goods and services. If too much money has been created and prices are rising, then taxation will bleed that excess money from the system. Bond issuance can serve that purpose as well--though because bonds pay interest, they are inherently inflationary, assuming that more money will exist in the future to pay them back.
In any event, neither taxes nor bonds can fund government spending. The US Government is the only legitimate source of dollars. It would be impossible for the government to demand dollars in taxes or borrow them with bonds, without them having been given to the people in the first place--usually in exchange for goods and services. Doing it the other way around would require universal counterfeiting!
Within this framework, MMT says that if maximizing economic output is an accepted goal, then it is desirable to fund investment into fully utilizing and expanding the productive capacity of the country, and this can be achieved by a fiscal policy of government spending to utilize slack resources that were otherwise not being used. The prototypical example is a jobs guarantee--if the economy can't figure out how to use unemployed workers, the government can be an employer of last resort, hiring them to do something or another that has some social benefit.
Again, I want to stress that MMT is not a set of policy prescriptions or economic goals. It's a description of how public finance actually works, and a set of economic and fiscal recommendations conditionally put forward based on state policy goals. Those recommendations are not ends in themselves, but are in service to political goals set by the political process (contrasting with, say, favoring deflation is pretty much baked into hard-money metals-standard economics).
So let's leave MMT out of the rest of this.
Japan & the Deflationary '90s
This gets to the point: the question of running inflationary monetary policy is one of political goals and political decisions. By now, Japan no longer has a deflation problem. They do have positive inflation rates--could be higher, but reasonable. The issue was more in the 90s to early 2010s. There are two components of this.
First is the larger global context. In the 1990s, the massive revolution in personal computer technology led to a sustained investment boom across the industrialized world. Those investments really were investments--they increased per-worker output in nearly every industry they touched. This was happening just as a wave of "Third-Way" politicians were also gaining prominence--people like Bill Clinton in the US and Tony Blair in the UK, who wanted to stake out a position of being pro-business but socially liberal. They took advantage of the computerization boom to draw down support for public spending--which, after all (or so the thinking went), wasn't needed; everybody was flush!--and use the dividends to continue reducing taxes and reducing regulation while also paying down government debt. This was an inherently deflationary goal, but it was a widespread one at the time. That's the global context for Japan's deflationary decades.
In the more Japanese context, the 90s and early 2000s were coming off a sustained economic boom from the 1970s-1980s. The investment boom ongoing in the US and Europe had hit Japan a decade or two earlier. A tremendous amount of investment to rebuild the infrastructure destroyed in WWII--much of it coming from outright gifted foreign aid--had led to the Japanese Economic Miracle, which was still paying huge dividends through the 1980s. However, towards the end of that period, most of the sustained growth was starting to die out: the obvious avenues for infrastructural improvements and productive investment had been pursued already. By the late 1980s, Japanese companies--which had always been (and remain) very well integrated with the government, even in the immediate pre-war period--had a ton of cash and no real good ideas for how to use it usefully. This was the period in which Western commentators were terrified of Japanese money, as Japanese businesses, for want of a better idea, bought American companies and American real estate (Pebble Beach, for instance). That's one of two ways accumulated money can go--it can either pursue productive investment, or buy scarce goods and live off rent.
Well, the result of all that success was a bubble, and the bubble popped spectacularly in 1989. The major corporations--all tightly integrated with each other, with the Japanese government, and with the banking industry--suddenly had deeply-underwater assets, many held overseas. The government did actually shift policy gears to try to keep them afloat--there were repeated bailouts of Japanese banks, for instance, and a concerted effort to lower the value of the yen relative to other currencies in order to stimulate exports. Unfortunately, the popular reaction to the bust was to become very conservative about spending money. The public demand switched from consumption to reserves--savings--which still had no particularly useful investment outlet. Now you have a vicious cycle where the common people all try to save. If you raise inflation too much, you'll only drive people to save harder, tamping down the emerging inflationary trends. Japan would have been inflating away the yen-denominated deposits of its citizens and assets of its banks, while the banks' attached companies kept paying for foreign investments made in other currencies. As a result, the government and Bank of Japan took a cautious policy approach that lasted for nearly twenty years. Couple that with the fact that the Rest of the World was pursuing a deflationary, debt-paydown policy at the same time--meaning Japan could not benefit from the extra demand created by government spending overseas--and they were, frankly, stuck, without the political will to increase spending in productive ways, nor to produce enough inflation to actually restart the economy.