Can children on a Thai DTV visa enroll in schools while in Thailand?

It's very simple and makes sense. The DTV is for a course the DTV holders will follow or other specified purpose. If they or their dependents join a formal school or a university, they must have a visa from the academic institution. It has always been like this

Where precisely does it say:

“A consent letter from your parent to acknowledge and understand that holders of DTV including principal DTV holder and all dependants are not permitted to undertake or participate in formal education programmes in Thailand under this visa category.”

Was on the Thai E–Visa PDF File itself??

Which (where) Thai Embassy/Thai Consulate is this??

Was this a direct request from the Thai Embassy/Thai Consulate to upload additional documentation on the Thai eVisa Website or an email directly from the Thai Embassy/Thai Consulate??

Was this on the Thai Embassy/Thai Consulate Website or Facebook Page??

Maybe they mean they aren't eligible for Thai public school?

That literally makes no sense. The DTV has no prohibition from attending school while you are in Thailand. And studying in Thailand does not require an education visa, you can study on any visa. I would just ignore it. The embassy doesn't get to dictate their own rules once you are inside of Thailand, it is immigration that controls things inside of Thailand.

The only issue you may run into is that some schools require the students to be on an education visa sponsored by the school. This is because they want your child's legal status in Thailand to be dependent on the school and you continuing to pay tuition. They feel it gives them more power and leverage over you, that if you try to leave the school, your child no longer has permission to stay in Thailand. I would avoid any school that requires this, and find a school that simply requires your student to be on legal status inside of Thailand.