No offense, but that's just silly. The president is the chief executive not the chief justice. We already have a Supreme Court. In fact, I think the idea of a president unilaterally "validating" (whatever that means) the constitutionality of a bill itself violates the separation of powers specified in the Constitution.
Besides, exactly how would the president "investigate, explore and validate"? By commission or all by himself? And what happens when his decisions are disputed? They'd only wind up in the Supreme Court anyway.
Yes and no. We have a Chief Justice and supreme court, but they’ve slept through most of the 20th century and appear to be sleeping soundly through the present nightmarish travesty on Capitol hill. The purpose of roping ol’ POTUS into the mix is to underline the antiquated concept of checks and balances, to emphasize that the Executive Branch has a co-equal responsibility to ensure the constitutionality of any legislation signed without passing the buck to the Court and, above all, to promote all the gridlock we can achieve to prevent any actual governance from occurring. Jefferson noted, “The two enemies of the people are criminals and government”. While I agree, I’ve always thought he was being redundant.
To answer your specific question, the president should “investigate, explore and validate” “all by himself”. It doesn’t take that long to read the Constitution-—the only place to look for constitutionality. The idea is to compel each branch to derive their just powers once more from the consent of the governed as enumerated in the document they refuse to otherwise consult.
If the President is too busy to verify the constitutionality of a bill before he signs it, he’s too busy. Passing bills too long to read, signing laws without bothering to check their legality and outsourcing responsibility to the snoozing Court has gotten us this far. Yes, it would (hopefully) wind up in the Supreme Court, as you say. Which is why it’s unnecessary to require that in the amendment. That achieves the aim of bringing all three branches face-to-face with the law of the land.