If you pay it from government coffers, then the government has less to spend elsewhere.
Not true in the US (and a few other countries). The US has economic sovereignty. This means that the federal government primarily owes it’s debt to itself, and only a very small percentage is owed to other countries. The fed also relies on a fiat currency, meaning that money has value because the federal government says it does. These 2 facts mean that the only limit to US spending is the amount of labor and resources available to the government at any given moment (please note that this is not true for state/local government). Haven’t you ever wondered why we have unlimited funds for the military but it’s austerity for everything else? It’s because conservatives in government want to hide these facts to continue pushing their agenda.