It is true that Howe wasn't pursuing victory very doggedly at the beginning of the war. He was looking forward to a negotiated peace and thought that if he crushed Washington, it would only embitter the Yankees and prolong the war.
But if Britain didn't crush Washington, odds are it was because they couldn't. Think Vietnam. We had the firepower and might, but in the end, we still didn't win. Plus, the French, the Dutch, and the Spanish all joined our side and Britain would have a lot of trouble defeating all those foes.
Had King George III decided to keep sending armies against Washington, he could have done so. He relented.
He wasn't an absolute ruler. He had to cope with Parliament.
Six or eight years and George III still didn't win the war. Parliament was getting restless. So he gave up.
If the Confederacy had been able to stick around for another two or four years, Lincoln probably would have had to give up as well.
you might mention that through most of that six or seven years His Most Britannic Majesty was at war with the King of France. Said King, possessed a very large and capable army and a large and capable Navy. With considerable more resources to fight a war that was available to the Colonials.
Admittedly, Lincoln’s logistics were far easier than George’s.