Any organization that wants to use a supplier ISO 9001 certificate as a component of the supplier approval & oversight process has to be prepared to engage with the supplier's CB, if and when the supplier shows not to be serious and sub-standard.
I would greatly appreciate improved CBs and greater accountability of CBs for their certification decisions. Engaging with suppliers' CBs seems like a good idea, especially if other organizations would do it (and I wasn't the supplier involved), but it's not likely to become common in the real world.
The problem is that engaging with the supplier's CB usually doesn't help the organization, except for maybe giving them remote hope that this will somehow drive the CB to improvement. Many of us who have dealt with CBs are cynical enough that we don't get our hopes up that they will improve.
If my supplier's nonconformance is small enough that I want the supplier to fix it and continue working with them, I'll work with them directly to get exactly the fix I want, and won't annoy them by involving a CB (who could overreact and suspend their certification and/or demand non-value-added corrective actions, both of which could hurt both the supplier and my organization).
Engaging with the supplier's CB will probably destroy the organization's relationship with the supplier, so I would not encourage my organization to do this unless we were absolutely sure that we never want to use this supplier again. Even then, I'd be careful depending on the power dynamics and risks of having other suppliers refuse to work with me because I reported one of them to a CB. I would have to expend effort/time/money to report the CB and I risk blowback for making the report; the remote chance at CB improvement is not a benefit that is worth these costs/risks.