I grew up in New Orleans and now live in Seattle, and all of my intuition for pronunciation is just totally broken.
In many places, if you see a "weird" (as in non-English-seeming) place name, a reasonable pronunciation guess is to just assume it's from a Romance language and pronounce it vaguely Spanish/French-ish. This works because so many foreign names that are common in the US that aren't obviously Anglo/Germanic are from Spanish, French, or Italian immigrants.
New Orleans screws that all up, though, because it has such a complex intertwined cultural history. In New Orleans, place names often have a pronunciation that is explicitly weirder and sort of the opposite of phonetic. You mentioned "Natchitoches", which is pronounced locally like "Nackodish". There is no reasonable algorithm that would take as input the spelling of a place name in New Orleans and output its pronunciation.
So my usual algorithm for pronouncing an unfamiliar name is, "Guess that it's like a Romance language and if not assume it's completely weird and unrelated to the spelling."
But Native American-derived names in the Pacific Northwest often confound that. They don't have Romance vowels or emphasis at all (for obvious reasons). And the pronunciation often is very close to the spelling. (I assume that's because the spelling came along so much more recently here in the PNW than on the East and Gulf Coast, and hasn't had as much time to drift.)
My dumb algorithm for pronouncing PNW placenames is "Imagine an American who's never even heard of a European country much less visited one, and have them pronounce the name phonetically." And it works surprisingly well!
For example, "Mukilteo". If you try to throw some Romance flair onto it, you'd get "Muh-KILL-tey-o", which isn't right (but does sound charmingly exotic). It's anybody's guess how that would be pronounced if it were a street in New Orleans. Maybe "Mill-toe".
But if you imagine some hopelessly bored midwestern kid forced to read it out loud in school and not even trying to get it right and they'd go, "Muh-kill-TEE-oh" and... that's it.
Likewise, I kept wanting "Anacortez" to sound like some Spanish explorer "Anna CorTEZ". But, no, it's just "Anna-CORE-tiss". "Humptulips" is literally "hump tulips". "Chimacum" is "chim-uh-cum". "Snoqualmie" is "snow-quall-me". They all have the most vanilla-sounding pronunciation.
I admit that Sequim ("skwim") and Puyallup ("pyoo-A-lup") are weird.