"Quartering Troops" sounds archaic, but every now and then lawmakers propose it, like forcing private airlines to provide discounted or free arrangements for servicemen traveling. (As opposed to negotiating rates with them, or soliciting competitive bids). These proposals usually don't get very far, but it shows you that the Government still thinks it's OK to _force_ a citizen or private business to give free services or accommodations to soldiers.
Disregarding what the law says, the material reality is that American society today largely respects and thinks highly of the US military, and is not as skeptical of federal government's standing armies.
This, by the way, tells you a great deal about how far away the modern American society is from its own founding principles - and especially so among the people who otherwise profess their infatuation with them (and who tend to be more pro-military and pro-police).
I think the idea is that directly quartering troops is certain to be much more capricious and unequal than paying taxes. Everyone (with some narrow exceptions) is supposed to pay taxes, while only those families who have nice houses in convenient locations (and beautiful teenage daughters!) will have to quarter troops directly.
(For example, see https://www.huffingtonpost.com/2011/06/08/congressman-bruce-... and https://www.huffingtonpost.com/2011/06/08/delta-troops-afgha... The Government should reimburse him, and try to negotiate the best rates from the Airlines. I'd be all for the Airlines offering a discount, but they shouldn't be forced to.)