I actually didn’t know that the policy was originally based on that reading of the Constitution; that’s pretty interesting. But it does seem like that reading goes too far. The OLC guidance is that a sitting President cannot be indicted, and the Constitution doesn’t say that. At best, it leaves open the loophole that if a President were to be indicted, they could easily pardon themselves. At which point any functional Congress would indeed move on to impeachment, for obstruction of justice. But if we let it happen that way, we’d probably lose the chance to prosecute the President for the original crime, so clearly it’s better to impeach first and then indict.
I can’t say for sure whether your interpretation is a conventional one, because I know very little about Constitutional law. You said it might not be popular, so I’m guessing there are other interpretations out there. But if you turned out to be right, the obvious moral obligation for any honest president would be to work toward a new Constitutional amendment to close that loophole. I’d still want to see any presidential candidate commit to that version of the pledge as well.