Absolutely correct.
I’d quibble with this list a bit. You first pointed out some of the schools in the elite tier that TCU doesn’t compete with. There’s absolutely an elite tier that isn’t on our radar. The Ivies and near-Ivies like Stanford, Rice, Duke, etc.
Then there’s a sort of second-tier. Your Vanderbilts, USCs, Notre Dames, Emorys, UCLAs, Michigans, etc. Objectively more rigorous admissions statistics and more national brands. These are our “aspirant” schools - or who we want to be. TCU competes with them for some students and tries to peel them away by offering huge scholarships. For example, a kid from Austin may apply to Vandy and TCU, get into both, and get a full or 3/4 ride to TCU compared to a 1/4 or 1/2 ride to Vandy. This probably works to TCU’s favor more than you might think. Especially for kids from the region.
TCU classifies its peer schools as SMU, Baylor, and Tulane (they actually cite Vandy, too, but I think that’s a reach). These 4 pull a large part of their student bodies from a very similar applicant pool.
What’s unique about private schools is they are pulling more from the extremes of their applicant pool. This is a feature of the private school enrollment model. When your sticker price is high and endowment massive, you use a big scholarship budget to pull some students from elite schools and let some further down the applicant food chain pay full freight if they want to come. That’s how you get a class with kids who turned down Vandy or Duke with no scholarship for a full ride to TCU, and kids who didn’t get into UT or A&M and didn’t get any scholarship offers from lower-ranked schools like OU or Tech but are still able to pay $50k/year+.
We aren’t competing with LSU. Sure, there are applicants that consider both. But TCU isn’t going to lose sleep over many applicants who choose LSU. Same goes for OU. Find me a TCU student who wouldn’t be an auto-admit to either of those schools.
Where TCU and SMU, Baylor, Tulane-type private schools miss out are at the level of very good—but not exceptional—non-upper class applicants/students. There’s a huge chunk of kids who can get in to TCU but just can’t afford it. And they often end up at a public school from UT or A&M to OU, LSU, or Tech if they aren’t competing for substantial TCU scholarship money.