I don't know about not being able to find a puppy. Petfinder just returned 367 matches to a search of Poodle-Saskatchewan-Baby. I imagine it would return a whole lot more for a location with more than population 1,000,000.
The issue with using Petfinder stats is that they are not very reliable because dogs in shelters are very often misidentified. Also when you view matches on a search, they are not only in your immediate area. For example, when I search Belgian Shepherd for my zip code I get 300 matches. Page one is dogs in NE Ohio or W PA but by page 4 (matches 76-100) I'm getting dogs in Canada, SC, CT and GA and by page 12 (matches 276-300) the dogs are in CA and AK. Of those dogs, a quick glance at the pages shows that very, very few of them are Belgians or even Belgian mixes.
On the same note, when I search for "baby" Standard Poodle matches, I get 40 matches with 2 pages of results. The results on the top fo the first page are in PA but the bottom of the page is already showing matches in NC and the second page is matches in GA, WI, AR, AL MO and CA.
Those matches include:
2 nearly grown purebred standard, certainly not a baby puppies
1 is a 10 month old Labradoodle - again not a baby puppy
6 purebred standards from different groups all over the place, some with adoption fees as high as $500
2 littermate Labradoodles, no age given but again not baby puppies with $400+ adoption fees
1 puppy guessed as a Weimerdoodle
1 adult toy Poodle mix mislabeled as a Standard Poodle mix baby
1 guessed as Goldendoodle/Husky/GSd mixes that looks like a spaniel of some sort
2 shorthaired litters that the shelters are claiming are Labradoodles
1 litter of shorthaired, short eared black puppies the shelter claims are "coondoodles"
1 adopted 4 month old labradoodle puppy
2 accidental whippet/poodle mixes
1 litter of Great Dane mixed with a standard poodle mix
Of these dogs, only 4 of them are even somewhat local to me - the Great Dane mixes.