If not, is there ample evidence to support this idea?
That's in essence the thesis of Jon Latimer's 1812: The War with America, in which he states unequivocally that the War of 1812 was a "failed war of expansion."
Latimer's work is one of only two academic treatments of the entire conflict - the other being Donald Hickey's 1812: A Forgotten Conflict - and takes a rather less optimistic view of American goals. Hickey's work is, while not overly nationalistic, much more sympathetic to the American cause.
I tend to favor Latimer's take than Hickey's, personally, as rapid expansion was a major element of the Jeffersonian-Madisonian domestic policy starting in 1800 onward. The Adams administration, on the other hand, was much less interested in expanding American territory, and had in fact rather aggressively policed the borders to evict squatters from treaty territory - which was, depending on who you ask, either calculated to reduce the possibility of violent encounters with native Americans or to bolster the speculation market. Regardless, Jefferson was keenly interested in acreage, because he championed a kind of American policy that connected land ownership to political enfranchisement, and therefore empowerment. Optimistic views of this goal argue, more or less, that Jefferson wanted a republic of yeoman farmers, and the only way to do that was to ensure everyone had access to affordable land, which meant that the country necessarily had to expand.
The tricky part was that the land necessary to expand into was occupied, and this is where the less favorable evaluations of Jefferson's presidency come into play. The problem was not so much that Jefferson gave much thought to Indian sovereignty, but that the middle ground of the Great Lakes region and along the Mississippi brought powerful tribes and loose tribal affiliations into contact with a still very potent Great Britain. In the 1790s war with the Wabash Confederacy, Great Britain had supported the native side with arms and other supply and materiel, and a bellicose Anthony Wayne had come very close to sparking a shooting war with Britain at the walls of Fort Miami after the Battle of Fallen Timbers.
Britain backed off a step at that point, but their presence along the old northwestern frontier allowed later confederacies to use them as a political bulwark against the United States. Tenskwatawa's preaching, and Tecumseh's strategy in the years from around 1807 or so up to Tippecanoe and the US declaration of war against Great Britain in June of 1812, included using the reality of promises of British support to encourage action against the United States. While Britain was nothing much more than a tepid ally, the possibility of their support in open war was an important feature of the confederacy that challenged the United States.
Britain, until the declaration of war, was nothing much more than a tepid ally, in part because Britain had no interest in a large shooting war with the United States. They had their hands full with the wars of the French revolution, and eventually the coalition wars against Napoleon. So while US warhawks, whose rhetoric slotted neatly into the Jeffersonian expansionist trend, pounded the war drum against Britain and made mountains of the British-Native molehills, they also saw cause to brand Britain's (very real) interference with American trade, and especially Britain's impressment policies as overly harsh and especially punitive to the United States. The Chesapeake Affair in 1807 brought this into light, but was less about British impressment policies as it was that Britain violated American sovereignty by firing on an American warship. This is not to say that impressment of American, nor nominally American, sailors wasn't an issue, but there has been very little satisfactory evidence that impressment was as big an issue as many American leaders argued. The commonly cited figure is that British ships impressed more than 9,000 American sailors between at least 1795 and 1812, but that figure is difficult to prove. Nevertheless, it was one of the chief popular causes for support of the war, which was in no uncertain terms extremely unpopular. Even Hickey argues that it was America's least popular war.
It's not hard to see how a conflict with Great Britain involving free trade and sailors rights may have shaken out without the invasion of Canada: the Quasi-War of the Adams administration had shown an enthusiasm and skill within a very politically limited framework that saw its goals accomplished without recourse to an invasion (though Hamilton and others loudly pushed for it). Expeditions against the Barbary states also showed that the United States was more than capable of pursuing violent political objectives within a limited and collateral-damage conscious paradigm, under the direction of both major political parties. Britain had, in fact, in the months prior to the declaration of war, backed off quite a bit in their trade war and in their treatment of American ships, and while American rhetoric claimed that the seizure of Canada was a mere political bargaining chip (the capture itself being, of course, a "mere matter of marching"), it's hard to believe that the war would have happened if the trade war and impressment issues were the only ones on the table. They clearly weren't, and the stapling of Native sovereignty issues with supposedly punitive British political goals in North America made, I believe, the chief driving forces of the war, and made the seizure of Canada the overriding goal of the war. What to do with it afterward was anyone's guess, but suffice it to say it never became an issue, because the US never managed to secure virtually any British territory for any great length of time.
I have written extensively on the War of 1812 in the past. A few of the more notable posts include: