If the employer is not intending to make any contributions then a non-ERISA 403b is the ticket (no ADP testing, no 5500, yada, yada).
If the employer intends to make contributions that would meet a 401k safe harbor then 401k is the ticket (larger universe of investment/record keeping platforms, avoid universal availability, plan investments not restricted to annuities or mutual funds)
Do they do anything else? Like check the appropriateness of the contributions? make the employer certify they have no other employees, or that those employees do not qualify for the plan? Keep the documents up to date?
The big difference is the universal availability rule—403(b) plans have to cover everybody, immediately, with limited exceptions. A 401(k) plan on the other hand can have a service requirement. The flip side of that is a 401(k) plan is subject to the ADP test whereas a 403(b) is not.
I've never seen that specific approach, but at a high-level it sounds basically like the trendy LSA benefit that's had pretty widespread adoption in the past couple years. Here's an overview: https://www.theabdteam.com/blog/lifestyle-spending-account-compliance-considerations/
The language about prohibited mid-year changes applies to plans that are already safe harbor plans. It does not apply to a plan that does not have a safe harbor provision in place. The "(or add)" that you bolded is referring to modifying or adding a match formula to an existing safe harbor plan, not turning a non-safe harbor plan into a safe harbor match plan mid-year.
On your second comment, a plan that suspends safe harbor contributions mid-year is not a safe harbor plan for that year. They cannot re-add a safe harbor match mid-year to become a safe harbor plan again for the year.