The British had overwhelming forces in North America, but had not been able to do much, aside from burning Washington. It is obvious that the British were eager to get the war over with and get their army back to Europe. Otherwise they wouldn't have agreed to those terms.I don't know if anyone in the US proposed continuing the war after the Treaty of Ghent was signed.Ratification was unanimous since the terms were outstanding for the US. Notably that for the first time there was a legal commitment for all the trading companies located in Canada to recognize America's right to prevent trade of weapons with the tribes. The Western States whose representatives voted for the war had their victory with this recognition in Treaty of Ghent. (Article the Ninth...)
As discussed in other threads, Britain only got 25,000 soldiers to Waterloo, but had 50,000 in Canada and the Bahamas at that time. The British were worried about some sort of trouble with France, Russia, Prussia or whatever and wanted their army back. They also didn't want a pointless major war trying to defeat the US after 20 years of war with France.
The war was an annoyance to Britain all along. That is why some in the US thought it would be successful. The British had the forces available after the fall of Napoleon, but by late 1814 it was just an annoyance for them again.