Why Soviet cancellation weakened the case for battleships

Did cancelling Soviet battleships help convince the West that battleships were no longer needed?

Yes; it removed the opposing battleship that offered the clearest procurement justification for maintaining or building Western capital ships.

Aircraft and submarines might destroy a battleship, but procurement arguments could present another battleship as the most certain symmetrical answer. That political logic does not require other weapons to be ineffective; it depends on describing the matching platform as the least uncertain counter. Once the Soviet Union abandoned its own capital ships, the direct threat disappeared, taking with it the strongest political case for Western equivalents. The same style of threat-based reasoning recurs in debates over carriers, nuclear submarines, and satellites.