This might be related to the ol' chicken and egg question: Does a good COA always end up with a center-aligned regulator?
Let's say you were handed a "serviced" watch. If you timed it and found that it was running a little slow or fast, would you come to the conclusion that it a) needed a regulator adjustment, or b) really didn't get a good COA? I imagine that your conclusion might depend on the watch and the amount of time loss or gain, but what if it was several minutes/12 hours in a lever-set watch?

If it was several minutes in 12 hours, I would send it back to the servicer. There could be a problem as minor as hands rubbing on the dial but in any case, it represents a poor job and should be remedied at no further cost. If it's only a matter of seconds, there is nothing wrong with having to adjust the regulator a little. At least those are my opinions.

Ken, you pose a very important question.

WHEN we are LUCKY, many of the less abused watches can be made to work very, very well. Sadly some watches can only be brought back to "life" to run fairly well for the sake of all the abuse they have suffered at the hands of their previous owners and service persons.

So "COA" is a misnomer. Servicing a watch begins with a detailed dis-assembly. Then cleaning, full inspection of all components, free spin testing of all the moving parts and very careful micro lubrication of the pivots during re-assembly so they do not STICK! Then we work to bring the balance wheel rotation to as large an arc as possible under the circumstances of that watch's condition. Finally we work with the Balance Weights, Hairspring and regulator pins to refine the escapement rate to compensate for known errors in the mechanics. If the watchmaker chooses the accepted practice can be to (sometimes) use the regulator to "fine tune" the escapement rate within the first 1 or 2 marks from the center position. Any more than that(for example on an eBay advertised as "serviced") is cause for some doubt about the last service.

Do NOT try to adjust the regulator as all that usually does is screw up a (qualified) service person's work.

Depending on the watch (in general) the regulator on "factory fresh" US-made Pocket watches can be expected to "adjust" the time error from 3-4 minutes/day (US-Made low jewel watches) to less than 20 seconds a day (Higher jeweled watches). After 60-170 years "in the field" it is anybody's guess what the regulator can (or cannot) adjust. Only the watch maker who serviced it can tell you.

I have shipped watches to people in the middle of the winter who messed them up totally trying to get them to work immediately after unpacking their 0 deg to -20 deg. F. box. When you unpack a watch give it a rest time to "soak" to your temperature. Then try it.

I have serviced watches for people who complain about time accuracy of watches with an uncut wheel and/or flat hairspring. These watches have little to NO TEMPERATURE STABILITY AT ALL and WILL deviate as much as 20 minutes a day. All I can do with them is adjust their speed to room (72 deg F) , or "body" (98 deg F) temperature to slightly improve their accuracy.

In summary, I advise that ONLY AFTER you have paid someone to service the watch and IF the watch does not perform to your expectations, should you seek advice from that service person you paid.

Wow. That was a MUCH more thoughtful and careful answer than expected, David. Thank you for myself and everyone else who will learn from it.