I fail to see how implementation of such a process would really benefit Delta (or any airline).

Put more specifically, I fail to see how the lack of implementing such a process would be a deterrent to passengers. For example, there aren’t going to be many (if any?) passengers who say, “I will only fly United and not Delta, because (for example) United has a digital health passport and Delta doesn’t”. Basically, there is no benefit for the airlines to implement it.

Of course there may be certain nations who could require the airlines to offer it in order to serve their nation, and thus the ‘passport’ becomes an involuntary action imposed upon the airline. And thus the answer to the OP’s question is simply “whenever some nation Delta serves requires it”.