There are a lot of excellent examples of governments solving problems by printing money. Most notably, basically every case of hyperinflation ever has been solved by printing money. Noted economist Mark Blyth goes into this extensively in Austerity: The History of a Dangerous Idea.

What MMT posits is not that governments can freely print money with no consequences, but rather that we're looking at the question if printing versus borrowing versus taxation backwards.

The basic theory of inflation is that too much money chasing the same quantity of goods and services pushes up the prices of those goods and services in an effect rather like a bidding war.

What MMT points out is that from a government finance perspective, this implies that the chief concern is the amount of money in circulation. If money is removed from circulation, via taxation, that becomes a control on inflation. This allows a radical rethinking of government spending, since your chief concern is no longer the balance of the budget, it is only the balance of the economy.

This also gives you new tools with which to combat inflation. Right now, in contemporary liberal economics, the only control we have for inflation is interest rates, which disproportionately harm the least affluent. But by considering taxation as an inflation control we can shift the greatest burden of combating inflation onto the wealthiest instead.

Cory Doctorow gets into a really good and approachable explanation of this in one of his more recent articles, when pondering the question of whether it is beneficial or even possible to eliminate the national debt:

There is only one source of US dollars: the US Treasury (you can try and make your own dollars, but they’ll put you in prison for a long-ass time if they catch you.).

If dollars can only originate with the US government, then it follows that:

a) The US government doesn’t need our taxes to get US dollars (for the same reason Apple doesn’t need us to redeem our iTunes cards to get more iTunes gift codes);

b) All the dollars in circulation start with spending by the US government (taxes can’t be paid until dollars are first spent by their issuer, the US government); and

c) That spending must happen before anyone has been taxed, because the way dollars enter circulation is through spending.

You’ve probably heard people say, “Government spending isn’t like household spending.” That is obviously true: households are currency users while governments are currency issuers.

But the implications of this are very interesting.

First, the total dollars in circulation are:

a) All the dollars the government has ever spent into existence funding programs, transferring to the states, and paying its own employees, minus

b) All the dollars that the government has taxed away from us, and subsequently annihilated.

(Because governments spend money into existence and tax money out of existence.)

The net of dollars the government spends in a given year minus the dollars the government taxes out of existence that year is called “the national deficit.” The total of all those national deficits is called “the national debt.” All the dollars in circulation today are the result of this national debt. If the US government didn’t have a debt, there would be no dollars in circulation.

The only way to eliminate the national debt is to tax every dollar in circulation out of existence. Because the national debt is “all the dollars the government has ever spent,” minus “all the dollars the government has ever taxed.” In accounting terms, “The US deficit is the public’s credit.”

https://doctorow.medium.com/retiring-the-us-debt-would-retire-the-us-dollar-30f366e0bc40

I'm citing Doctorow here rather than academic sources because he's very good at explaining this stuff in comprehensible ways. For a more extensive breakdown The Deficit Myth by Stephanie Kelton is regarded as an excellent resource. Kelton, BTW, was formerly Chief Economist on The US Budgetary Committee; she absolutely knows her stuff when it comes to government level finance.