I feel like it is mathematically impossible for this strategy to be good

No you are absolutely right it's not a good strategy. But that's not what the problem is. The problem is how do you fix that unsustainable situation.

If you try to reduce your budget deficit fast during a downturn then you may still not be able to reduce your debt burden, since reducing government spending also reduces economic output. The assumption that the Greek rescue was based on was that this reduction in GDP would be smaller than their saving in government spending. For example you reduce your budget by 1 million and output falls 900,000 then you are actually improving your situation. They believed that companies would see that the budget is under control and therefore start to invest more - restoring confidence is what they called that.

It's a phantasy. When you try to slash your budget during a recession then companies lack both orders from private firms and the government and will reduce their investment and fire people. So the result is that for every million "saved" in government spending you lose more than one million in output and your debt actually goes up.

Now, souverain countries that have their own currency have an easy way out of this dilemma. They devalue their currency. Inflation increases, imports fall exports grow. The government debt (as long as they borrow in their own currency) falls due to inflation and the necessary adjustment is much smaller and the devaluation makes domestic industries much more competitive. What you get is a short sharp crisis, and the IMF typically stands by with liquidity. If they have too much debt in a foreign currency they will likely have to restructure some of it, which happens all the time. Currently in Ukrain for example.

Greece doesn't have that option since they don't have their own currency. So the more they "save" the worse their economy and their debt become. It's a vicious cycle. The only way out is to write of a large chunk of the debt they owe and end austerity, which is selfdefeating when you lose more than one euro for every euro you reduce the budget.

So, the only hope for Greece is that a) a lot of the debt is forgiven and b) they aren't forced into more austerity, that will reduce their GDP even further. There is no way around it. It would be a very difficult sell in Germany though; and there are worries that if other countries see that Greek debt is forgiven then they can pile on debt too since it will be written off. But Greece has now suffered through 5 years of depression it is not very likely that any country would choose to go down that path.

Basically, I am worried my country will be another Greece in 20 years.

Does your country control and borrow in its own currency? If so then it cannot turn into another Greece. If it controls its currency but borrows in dollars then it's more likely that it will be another Argentina once a crisis hits. If you are Italian, well, yeah you are fucked and you will very likely face a very similar fate as Greece.

Devaluing is a get out of a terrible situation without too much damage card. While countries should definitely not borrow more and ore money during good times, as long as they do so in their own currency then they will not suffer the same fate as Greece. Inflation might wipe out a lot of your personal savings, so while a country is growing the government should try to build a buffer or at least reduce the debt. In a recession a country should spend more than they earn since the reverse to the above scenario is also true. Every dollar more spend will increase output by more than one dollar. So the country will get out of the recession faster. Grow again and then should use the next business cycle to pay off its debt (or at least have a deficit below inflation so the debt to GDP ratio slowly falls). So they again have the firepower to overcome the next recession.

But, that's sadly also just a phantasy. Most countries tend to spend all available money and more even during good times. Luckily, the damage caused by this remains limited as long as they have their own currency and also borrow in it.