Taking on debt means you can buy things now, and pay for them later. Tanks, planes, schools, hospital, paychecks of your employees... Very nice things, maybe things you can't afford right now. But if you take a loan, you can deal with it later. Eventually you will have to pay up, but by then maybe you can take a new loan, or maybe the economy will improve. Most likely you personally won't be around, and someone else will be in charge of the country. But you will get all the credit for the schools, hospitals, paychecks and tanks that your successor has to foot the bill for. Who knows, maybe the country will collapse by then or go bankrupt, and nobody will have to pay the debt after all.
This is what you lose: The ability to buy things. It is very similar to always stashing half your money in a hole and only letting yourself spend the other half. What are the downsides to spending less money? Well, you get fewer nice things. It's the same for countries.
Sure, if you use debt to pay for things, there will be interest, but then again the fact that you get to buy some thing now, rather than later, also represents some time-value which may or may not outweigh the interest. Paying your army this month so they don't revolt sure sounds better than having to pay an extra 10% or whatever in interest, doesn't it?
Debt is basically a source of capital. It can provide tremendous amounts of money, which allow countries to do all sorts of useful things. This is why in some situations debt is called "leverage". Yes, you need to service the debt (interest, etc.) but the servicing is not necessarily exorbitant, and the gain from obtaining capital is often greater. In fact, interest rates are adjusted such that loans are not a ripoff, otherwise nobody would be taking loans and the creditors would not be able to make money.
From the myopic perspective of an individual, being debt-free sounds something desirable
You haven't adequately justified this perspective. In fact, in many cases it is to the individuals advantage to take on debt, even if they have money. If a credit card offers 0% introductory APR, you're better off buying everything on that card without paying it, and investing the money for a year. If a dealership offers you a 0% 6-year loan on a car you want, it's better to buy it on credit and again, invest the money, or keep it around for an emergency. Often, you are far better off getting a mortgage rather than trying to save up hundreds of thousands while also paying rent.
If the individual is extremely disorganized, and forgets to pay bills, doesn't care to take 5 minutes to set up autopay for his car loan, does not understand that credit must be paid back and maxes out his $50k credit card when his income is only $3k/mo, of course this individual should not take on debt. The individual would be less financially efficient with this debt-free lifestyle, but it will minimize the risk from their own lack of organization. Countries typically do not have such pervasive inability to remember paying the bills.
If the individual is unable to plan ahead, and the thought of paying some large some several years later fills him with dread, even though he could easily save a small part of it every month, because he lacks the ability to plan long term, this individual should not take loans - they will generate needless stress and the additional funds won't be worth it. But countries also do't often have such problems with stress or planning.
Moreover, countries, unlike individuals, can always print money (although that devalues their currency and has pretty much the same effect as taking a loan). Countries, unlike individuals, can unilaterally void their obligations by leveraging their military or diplomatic power.
Also, often loans between countries come with strings attached, and the loan-giver gets some amount of control over the taker. A lot of countries like this sort of thing, as it provides them with yet another means of dominating smaller countries without having to do more aggressive actions. The smaller country can in turn feel safe from being seen as a threat, given the financial relationships. A country which goes out of its way to refuse taking on debt, can be seen as one which is too independent, and should be suppressed through military means. So for countries, unlike individuals, being debt-free has an additional downside of making them look suspicious and dangerous.