I understand what you are saying about the money issue.
Long ago when the dinosaurs roamed the earth and I was in HS we did not pay the fee for the AP test, but now kids do. In the 2 hs my kids attended in 2 different states it was you pay for the AP test the 1st week of school, if the child scores a 3 or higher they would refund you in June after they took the test. No for some paying is not a big deal, but when you have a child taking 5 APs that 1st week of school can be several hundred dollars. On top of that if the school has a Jump start program at the local CC you have to pay for their classes there too. One yr it was almost a 1000 bucks for us between the AP and the Jump Start classes.
That of course is before paying for the SAT and ACT all spring and fall or the 75 buck a pop for college applications. That 6 months between spring jr yr and fall senior yr is a great way to start prepping parents for how expensive college will be in tiny little steps.
I quickly came to realize that by doing this the schools were finding a unique way to reduce class size or hire less teachers and on top of that add to their discretionary funding.
That being said all 3 of our kids in hs were rounded up for a morning of PSAT testing and we did not pay for the test. There was no registering for the PSAT. The school registered all of the kids their jr. yr. To tell you the truth, I only found out about at dinner time...so how was school today? Fine, we took the PSAT and so now I have no homework! Seriously, that was how it worked both in VA and NC for all 3 of my kids. One did tell me the night before because he was scrambling to find his calculator!
I forgot that typically it is the kids on the coast, especially the east coast that really take the SAT. In our area of VA you can't risk the chance of paying a late fee if you want to take the SAT within a 45 mile radii, they book out that quickly. However, you can pass the late fee, and do a walk in for the ACT at the nearest HS because that is how rare it is for kids to take the ACT.
For that reason I change my statement about why a school wouldn't offer it. It maybe as you stated there is no real demand to have the school pay to administer the PSAT.
I would only stress taking it for one real reason. That NMSF/NMF issue. Look at the colleges you want to attend, somewhere in their big website they will have an academic breakdown of the students attending. They love to boast about the % of NMSF/NMF students. It is not a hook unless you are an NMF, but it is an edge even as an NMSF for admission purposes.
I would also take it because some kids do better on the SAT and some do better on the ACT. The rule of thumb is add a 0 behind your PSAT score and that is your SAT ball park. For example 209 would be 2090. You now can take the ACT in Oct/Nov/Dec. and work your way backwards by using the conversion chart to see which test is a better fit after you take the ACT.