Only the director has the right to award a 20H1 claim; Black has no right to award the claim onto himself. If as a player, if I was going to ask a director on a rule 20H1, with the other players clock running. I would let the clock run, as the director can reject my claim.
If as a director, if someone makes a claim on rule 20H1, would tell the player to let the clock run, then go look for this missing person. Would check the bathrooms, the outside of the building to see if the player is smoking, if I know the persons car would check the parking lot. Not going to spend all the time looking for this person. If the event is huge, I could find myself not able to look for the player. If the player cannot be found, will look at the board and the time left to make the final call. As a rule 20H1 could expell the player from the event.
The other problem for Black with the 20H1 rule. Black came back and did not see White, then stopped the clock. White could have been away from the board only a minute. If Black was at the board without White for 15 minutes, then Black has a better claim for a rule 20H1. If Black came back and did not see White, then stopped the clock to see the director, would reject the claim. As Black cannot prove White was away from the board for 15 minutes. The claim on the clock would not prove nothing, as a clock cannot prove how long it has been running on each move. Even if it did, White could have been at the board during that time.
If Black did make the claim, would have told Black to restart Whites clock and stay at the board. Would have gone to find White, and we know White did come back to the board. If Black did make a claim of rule 20H1, Black would have been at the board when the claim was being reviewed.
Black was in error, as he stopped the clock and left the board.