By the logic in that post, evidently we fought World War II because we wanted to annex North Africa and Normandy.
The U.S. invaded those areas to get at the enemy that had attacked us.
In 1812, perhaps the U.S. should have attacked the 300 Ships of the line and 300 frgates and 100 sloops of the Royal Navy with our 300 ships of the line and 300 frigates and 100 sloops - but all we had was six frigates. We did what we could; and it was barely enough to save the country within it's 1812 borders.
It was arguably the most glorious time in our history, and left-leaning America-bashing history professors delight in denigrating it.
In fact, the US maintained a major airforce base (Wheeler) in Libya for 28 years after the 1942 invasion, and still today maintains "forward-operating bases" in Morocco and Tunisia, among other countries world-wide.
In Europe we still have dozens of bases in places like Ramstein, Germany and Rota, Spain.
All are the result of treaties and alliances, some of which put our militaries under single command.
Of course, none of these will ever lead to annexation, but Canada in 1812 is a different subject.
Again, by my count there were nine invasions in less than two years, nearly all defeated.
But had they succeeded, why would permanent military bases, treaties and eventual annexation not have followed?
Finally, let me appeal to an authority on this subject, former President Thomas Jefferson:
So, with British forces expelled, why would Americans ever want to "give Canada back" -- back to whom?
Canada would become just another territory, like the Northwest Territories and Louisiana Purchase.