As you pointed out, the code does not address how continuity is verified. There is no reference to a continuity log, time cards, etc. So, it is in your lap and it is up to you and your employer to make a decision as to how you are going to move forward.
As you mentioned, it is easier enough with new hires, they are tested, the results documented, and if you decide to use a formal continuity record, add them to the roster. It is a clean approach.
What to do with employees that have been arcing and sparking for the last century? My question is, what are you fabricating, what type of inspections records are available, and are you willing to put your name and reputation on the line attesting to the fact they have been welding and producing acceptable welds? A big question is, do you have their qualification records available?
There is nothing wrong with or questionable about instituting a new system. Rarely, if ever, is the new system going to correct the omissions, sins, or errors of the past. If you are satisfied the welders currently employed are producing acceptable work, then simply add them to the continuity roster and move forward. If there is a question of the welder's ability, requalify the welder and move forward. The sticky point is you must treat everyone fairly and without bias. One way do that is to have everyone requalify on a 3/8-inch plate. That "renews" the welder's qualifications to what they were originally qualified to do and you have a clean system that is above reproach.
Best regards - Al