1. It's not smart to have a legal limit on a country's debt at all, no other countries that I know of have such a limit. Perhaps in the US it's a constitutional requirement, but in that event I would be rushing to get the constitution amended. If there has to be a limit, why not just multiply it by ten so no one has to think about it for a few decades, if ever! In a sense the current crisis is a purely artificial, technical one, since the US clearly has the economic capacity to service a much larger debt. The country does not face a liquidity crisis and is a million miles from being insolvent.
2. The US federal government currently has the lowest tax take of any advanced country, around 15-16 % of GDP. But its spending is about 25% of GDP, hence the current deficit of around 10% of GDP. With such low tax rates, it's got to be easy to balance the books with quite modest tax hikes - but many leading US politicians are dead against such an obvious solution. If taxes were already high, that would be understandable, but they're not.
3. Point 2 means that if the US gets to a formal default, federal spending has to fall - instantly, because the country won't be legally able to borrow - to be in line with tax revenues. Thus federal spending would have to fall by 40% overnight, and if the government decided to use some revenues to service existing debt, federal spending on services to the American people will fall even more, say by 50%. That would not be politically popular even if the problem only lasted a few days. It raises the question, who would be blamed by the voters? My heart sinks when I see the silly and dangerous games about this that US politicians are currently playing.
4. Last, if the US does get to a position of default, the international reputation of the country will sink like a stone and huge damage will be done to the sense of trust and reliability associated with US assets in world financial circles. More concretely, because US creditors will not have the same trust in the US as they have had hitherto, they will assign a non-zero probability of default to US assets in the future. Hence the cost of servicing the existing sovereign debt - and any future increases in it - will rise. It seems to me that some leading US politicians are nowhere near as scared as they ought to be about prospect of a default!