So you're saying that if a school doesn't meet some particular preconceived standard, Congress can prohibit states from recognizing its diplomas? That's not what that article states. The purpose of the article is to prevent states from disregarding the acts of other states, not to empower Congress to force states to disregard those acts.
Not so much that Congress would prohibit a state from recognizing a diploma as the reverse requiring a state to recognize a diploma from another state, as the article of the constitution expresses. I am not in anyway saying that such rejections are currently occurring or that they will, but the fact that they can, empowers Congress.
Implementation methodology should be a local or state issue depending on the source of funding. Even the choice of merely meeting or exceeding the minimum standards is a local or state issue. Establishing minimum standards is validly a federal issue because the document (diploma) is expected to be recognized on a national level.
An analogous example would be licensing. Each state currently tests for the CPA license. Passing the CPA test in Texas does not make you a CPA in all the 50 other states. Thus Texas can set the standards for CPA and congress is not empowered. If however the CPA in Texas was valid in all 50 states, Congress would be empowered to set minimum standards or methodology for the CPA license.
The high school diploma is issued by the state. If a job in Florida list a high school diploma or GED as a requirement, can they exclude someone from the job because their diploma was issued in Alaska? That addresses why Congress is constitutionally empowered to set standards. Just because they empowered does not mean the should or need to. However, whether I like it or not, Congress is going to spend a whole pot full of federal dollars on education. I want every single last one of those dollars accounted for. How else are you going to do it?