As I say, it would be nice if someone suggesting this as a possibility would spell out just what they think Americans of the time would see as the advantage to be gained by this extraordinary effort so extraordinarily far away from the regions of general American interest.
Now I've just gotten home from work, having taken time I didn't actually have (admittedly, just minutes) to find some indication of what an advocate of this course did say at the time. (Which by the way, in Shaler's case, was well after the War of 1812, though only a decade or so the British would still be extra wary of course!) So I still haven't read Shaler and frankly may never, as it seems like a pretty wacky, far-fetched sort of thing he's advocating. It's up to the OP here (and was up to danwild6 four years ago) to explain why anyone in the USA would even want this, and then to deal with the well-founded objections people raised both 4 years ago and now.
Shaler's idea seems to have been that since Yankee traders could not free-ride reliably on the protection of closer and greater European powers, Americans needed to have a base in the region for protecting their own shipping. Clearly for such a scheme to begin to have any traction in DC, the benefits to Americans would have to outweigh the estimated costs. Most Americans could care less if a clique of Yankee traders could sail safely to the Levant or not, and might suggest that these traders bear the costs themselves if they want the benefits.
In other words, the merchants could arm themselves, and if they needed a base of operations, they could bloody well filibuster one on their own hook--and take the chance that these private actions would get themselves labeled as pirates right along with the Barbary coasters.
What I recall of the earlier "millions for defense, not a cent for tribute!" brouhaha in the Jefferson Administration, from my general American history back in grade school (well, my extracurricular reading of it) was that generally the European powers did simply pay tribute to the Dey of Algiers; it was basically a protection racket, and apparently the British, French, Spanish and I guess everyone else reckoned that the bribes the Dey expected were cheaper than the cost of an invasion and trying to keep control of the place. I don't know to what extent they factored in trouble with the Sultan in Constantinople in counting those costs. But if the purpose of seizing control of Algiers would be to smooth the way for American shipping to the Eastern Med, the USA would not want to be on the Sublime Porte's bad list either, as the Ottomans controlled many of the ports Americans would want to trade with there (except those controlled by Western European powers, who presumably would not welcome Yankee competition there in any case).
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I took a few minutes to glance over Shaler's sixth chapter that I liked to above--there are a number of pages missing from this online version. Before the hiatus, however, Shaler was urging not the USA but Great Britain to seize control!
Perhaps the French read his rather glowing (and inflammatory, to the Ottomans--some diplomat this dude was, eh?) tributes to the potentials of the Algerian shore lands. Which by the way have been sold a bit short to be sure by some commentators here; clearly the Mediterranean lands of Algeria are a fairly decent tract of land, important to Carthage, to the Romans, and to the French who seem to have either taken Shaler's advice or figured it out on their own.
Anyway, despite coming to value Algeria to the point of declaring it an "integral" part of France, the French lost control of the place too. Shaler overstated the case when supposing the land would be more cost-effective for the British to hold than India or other possessions in America and elsewhere.
Anyway I'd have to read this whole book (which probably has other gaps in it, in this free online version) to get a sense of whether Shaler ever seriously considered the USA being the power to grab it or any part of it. Frankly I'd rather not.
So, Patton and danwild6, the ball is in your court, to try to make the case interesting. I think I've done enough, or too much, of your homework here.