I've been thinking about this along the same lines as you, and wondering if our goal should be less about knowing
how many there are, and more our
chances of finding one?! Not sure if this could work, but behold - the DC to find PC table
"DC to find PC"
Polity size_____Tier 1_____Tier 2_____Tier 3_____Epic+
100s___________10_________10_________20_______30
1000s__________5__________10_________15_______25
10,000s________5__________5__________15_______25
100,000s_______0__________5__________10_______20
1000,000s______0__________0__________10_______20
Of course the title is tongue-in-cheek, as the idea is we're finding character-class equivalent NPCs who will mostly be represented by abstract MM stat blocks. As a further caveat, I believe 5e DCs could meaningfully scale to 40. If you think so too, then this table might be improved using higher DCs for Tier 3 and Epic+. Could it work to introduce rolls to suggest headcount spreads e.g. d10xN where N is a factor based on polity size?
@
S'mon @
SkidAce
Heh, I think I would only make a DC check to locate something if I knew it existed.
In my Ghinarian Hills Wilderlands campaign the main, heavily detailed, campaign area is approx 2000 square miles, about 10000 people. Almost everyone is Tier I or II CR though there are a fair number of Priests similar to the MM one with CL5. Going through my NPC lists I could find two arguably Tier III - Oriax the Lecherous, CL 12? Wizard caster level 19 - currently on another plane, and Lady Meda, CL11? Bard caster level 15. I don't think there are any other casters in the region above about level 9 or 10, and warrior types top out in Tier II likewise.
Applying your numbers:
1% Tier I CR 1-4 would be 100 - looks about right.
0.1% Tier II would be 10 - hmm, I think there are probably slightly more Tier II, but mostly at the low end, around CR 5.
0.01% Tier III would be 1 - yes there is 1 Tier III in the region, though there used to be 2.
There are also two PCs who would be Tier III or IV
Shieldbiter Dragonborn Barbarian-15
Hakeem Barbarian-18
But they killed a bunch of Tier I-III NPCs to get where they are today.
Taking an hour to go over the entire region and guessing at a couple CRs for characters I've not statted yet, I get:
Tier II in region
Archermos, Gladiator CR 5
Joseph, Wiz 10 CR 6 (on another plane)
Sarene, Pirate CR 5-6
Damne, Weaponmaster CR 7
Jane Poole, adventurer CR 6
Lorius Vex, adventurer Wiz 10 CR 6
ex-Lord Hytirus Vex, CR 5
Lord Krens, Wiz 9 CR 6
Lord Kaldrac CR 5
Lord Usarus CR 5
Lord Arkonos CR 5?
High Priest Larsonos Clr 8 CR 5?
So 12 Tier II. 1 or 2 of these might be CR 4 but there might also be a couple CR 5+ I don't know about yet.

Given how rough my population estimate was, I'm struck by how closely your formula does fit this organically bottom-up detailed campaign area. If I took out the two who fled to an alternate plane it would be 11 Tier II and 1 Tier 3, almost exactly on. And other areas of my campaign world for which I have partial demographics seem to
have a similar distribution.
Re Tier IV - some PCs just woke a CR 20 ancient brass dragon at the bottom of an 11-level dungeon, and some other PCs are about to encounter an immortal lich-king in another dungeon, but I'll not count those.
Anyway my conclusion is you are definitely in the right ball park for a typical D&D campaign.
