Jelly, 18 bytes

J:g¥Ƭ’}ṪU×ḷb§¥ƬṪS%

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A less naive computation of the digital roots comes out a bit longer by having to special-case 0. Inverted output: truthy (nonzero) if the checksum fails.

J                     1-indices from the start
 :                    divided by
  g¥                  their GCDs with
     ’}               B - 1
    Ƭ  Ṫ              repeated while unique,
        U             reversed
         ×ḷ           times each N.
              ƬṪ      Loop while unique:
           b ¥        Convert each to base B
            §         and sum each.
                S     Sum
                 %    mod B.