To answer your hypothetical question the answer is the propeller in NOT airworthy. Why because it has to be airworthy and airworthy has two conditions as follows:
AIRWORTHY--- Two conditions that must be met for and aircraft is considered airworthy.
1. The aircraft must conform to its type certificate. Conformity to type design when the aircraft configuration and the components installed are consistent with the drawings, specifications, and other data i.e. STC field approved alterations.
2. The aircraft must be in condition for safe operation. Condition of the aircraft relative to wear and deterioration, i.e. skin corrosion, window delamination/crazing, fluid leaks, tire wear, vibration, etc.
NOTE: If one or both of these conditions are not met, the aircraft would be considered unairworthy.
First it is the pilot’s job to squawk things that don’t seem right. Now the mechanic is charged with determining if it is airworthy or not. It either is airworthy or its not.
So take the meaning above does the propeller met condition number 1. In this case let’s say yes it meets its time design of T/C.
Next condition for safe operation so lets take a look at the FAR rules. Under Part 35 propeller certification it states:
§ 35.4 Instructions for Continued Airworthiness (ICA).
The applicant must prepare Instructions for Continued Airworthiness in accordance with appendix A to this part that are acceptable to the Administrator.
Since you did not mention what hypothetical propeller we are talking about we can’t look up the ICA. However the manufacture has to perform vibration testing and set limits.
§ 35.37 Fatigue limit tests.
A fatigue evalution must be made and the fatigue limits determined for each metallic hub and blade, and each primary load carrying metal component of nonmetallic blades. The fatigue evaluation must include consideration of all reasonably foreseeable vibration load patterns. The fatigue limits must account for the permissible service deteriortion (such as nicks, grooves, galling, bearing wear, and variations in material properties).
The mechanic is charged with performing maintenance. Maintenance means inspection, overhaul, repair, preservation, and the replacement of parts, but excludes preventive maintenance. So the A&P or propeller repair station is required to follow FAR 43.13:
§ 43.13 Performance rules (general).
(b) Each person maintaining or altering, or performing preventive maintenance, shall do that work in such a manner and use materials of such a quality, that the condition of the aircraft, airframe, aircraft engine, propeller, or appliance worked on will be at least equal to its original or properly altered condition (with regard to aerodynamic function, structural strength, resistance to vibration and deterioration, and other qualities affecting airworthiness).
I happen to pull two examples of T/C’s for a Hartzell propeller the notes say:“(1) Airworthiness limitations, if any, are stated in Hartzell Manuals 108( ), 114( ) or Service Letter 61( ).”
McCauley T/C states; “Approved Vibration wise for Use on Normal Category Single Engine Tractor Aircraft. The maximum and minimum propeller diameters that can be used from a vibration standpoint are shown below. No reduction below the minimum diameter listed is permissible, since this figure includes diameter reduction allowable for repair purposes. “
So the question is what are the vibration limits in the manual or T/C did the propeller exceed them and if so the propeller is unairworthy and must not be returned to service. However the mechanic is required to make a maintenance record entry indication how they inspected it and to what data.
This is not a case where the owner can decide airworthiness. It’s the mechanic responsibility. The mechanic can reinstall it with a record entry indicating it is unairworthy in the maintenance record, because the owner operator is responsible for having the propeller repaired.
Propellers have inherent vibration characteristics, which are not usually harmful but can induce fatigue and in time cause failure of parts essential to the airworthiness of the aircraft. This is one reason why periodic inspection of the aircraft is essential.
A special word about propellers. Quite often a propeller blade becomes nicked, especially at the leading edges. These nicks become points of stress concentration. IT IS IMPORTANT THAT NICKS BE REMOVED AS SOON AS POSSIBLE AND IN A PROPER MANNER. Since the removal of nicks requires special skills and tools and a thorough knowledge of the procedure, such work may be accomplished by certificated personnel only. The importance of correct removal of even small nicks AS SOON AS POSSIBLE after incurring them, cannot be overstressed.
Below is some reference data that may help.
AC 20-66A Vibration and fatigue evaluation of airplane propellers
AC 23-16A Powerplant Guide For Certification Of Part 23 Airplanes And Airships
AC 20-106 Aircraft inspection for the general aviation aircraft owner
AC 33-2B Aircraft engine type certification handbook
AC 43.14-1B
8-109. VIBRATION. Although vibration can be caused by the propeller, there are numerous other possible sources of vibration which can make troubleshooting difficult.
a. If a propeller vibrates, whether due to balance, angle, or track problems, it typically vibrates, throughout the entire RPM range, although the intensity of the vibration may vary with the RPM. If a vibration occurs only at one particular RPM or within a limited RPM range (e.g. 2200-2350 RPM), the vibration is not normally a propeller problem but a problem with a poor engine/propeller match.
b. If a propeller vibration is suspected but cannot be positively determined, if possible, the ideal troubleshooting method is to temporarily replace the propeller with one which is known to be airworthy and test fly the aircraft.
c. There are numerous allowable tolerances in blade angles, balance, track, and blade width and thickness dimensions. These tolerances have been established through many years of experience. The degree to whìch these factors affect vibration is sometimes disputed and can involve significant repair bills, which may or may not cure a vibration problem. Reliance upon experienced, reputable propeller repair stations is the owner's best method of dealing with these problems.
d. Blade shake is not the source of vibration problems. Once the engine is running, centrifugal force holds the blades firmly (approximately 30-40,000 lbs.) against blade bearings.
e. Cabin vibration can sometimes be improved by reindexing the propeller to the crankshaft. The propeller can be removed, rotated 180º, and reinstalled.
f. The propeller spinner can be a contributing factor to an out-of-balance condition. An indication of this would be a noticeable spinner "wobble" while the engine is running. This condition is normally caused by inadequate shimming of the spinner front support or a cracked or deformed spinner
Just one man’s opinion.