So the district priests and leadership look into the case of the "N.O." priest, do whatever they do, and eventually assign the priest to a priory where he offers mass publicly, and assigns him a chapel where he also offers mass.
On what basis would a layperson, not privy to any of the details of the case, doubt this priest isn't a priest?
What is there to "look into"? The doubt fundamentally lies in the new rites. The "investigation" would take 5 minutes and is as simple as 2 questions:
a. "Did you get ordained in the new rite?"
b. "Was the bishop that ordained you, a bishop in the new rite?"
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If yes to either question, then there is a major doubt. If yes to both, then there are 2 major doubts.
Solution - conditional ordination.
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What other investigation can there be? The "feelings" of the candidate don't matter. The "orthodoxy" of the "bishop" or "priest" doesn't matter. What matters is the flawed new rites were used.