Perhaps the sensible thing would have been to create a completely new “presidential guard” to handle the job, but politicians and bureaucrats don’t always do what’s sensible. There may have been resistance to the idea of just creating a new agency from scratch on the grounds of not wanting to expand the federal government (especially not with yet another law enforcement agency, an especially touchy subject–there’s long-standing resistance in the U.S. to anything tantamount to a “national police force”). I also don’t know how ad hoc the assignment of presidential guard duties in 1901 was; we’d had previous assassinations as recently as Garfield in 1881, but somehow Lincoln’s and Garfield’s deaths hadn’t convinced anyone to do anything permanent in the way of presidential protection, and so in 1901 there may have been something of a scramble for a solution to the problem.
Given that they did give it to an existing agency:
The D.C. police department (founded in 1861) has never been a “national police force” or had, AFAIK, any jurisdiction outside of D.C. Granted, the Secret Service was an anti-counterfeiting agency, but at least it was an agency with a nation-wide scope.
The Park Police go back to 1791, but have only had jurisdiction outside of D.C.–in national parks, anyway–beginning in 1929, so the same objection as to the D.C. police would apply to them, plus, c’mon, they’re Park Police. All the other nations would make fun of us. (“Hey, what if he goes outside of a park, huh?”) (Actually, the Park Police have had full jurisdiction throughout D.C. since 1882, which, as I said, puts them in the same boat as the D.C. police.) At the very least, if they had been given the job, we’d probably be asking today “Hey, how come the Park Police guard the President and not the…”
In either an earlier or a later time the duty of guarding the President might have been given to the military; however, the current prohibition on the military taking on any domestic law enforcement duties (the “Posse Comitatus Act”) dates to 1878, and in 1901 I would guess that mentality would still have prevailed enough to scotch the notion of a military Presidential bodyguard. (According to that link, the military was pretty routinely used for domestic law and order purposes earlier in the Republic, and if we had somehow made it to the era of World Wars and Cold Wars without assigning anyone in particular the job of guarding the President, it might well have fallen to the military to do so. Earlier in our history, a peacetime standing army was unthinkable, and we generally had almost no foreign intelligence apparatus. A militarized presidential guard might have been instituted at the same time as the rest of the late 20th Century national security state, except we already had a civilian presidential guard by then.)
I think the U.S. Marshals Service (dating back to 1789) would have been the Secret Service’s most obvious competitor for the responsibility in 1901. (The earliest predecessor of the FBI only goes back to 1908.) Certainly the Marshals had a national presence and experience in a variety of duties. Pure speculation, but perhaps it was felt they already had enough to do, while the Secret Service of the era had the counterfeiting problem reasonably well in hand and was itching for something else to do. More seriously, bureaucratic turf wars and the desire to not lose potential missions (and funding, manpower allocations, and prestige) can result in tables of organization which don’t always seem to be the most objectively rational. Perhaps the Treasury Department of that era was just a bit better at infighting and lobbying Congress than was the Justice Department.