Hi Andrew,
The simplest solution is to publish products through an on-demand workflow, so that you can batch run workflow on multiple products to publish them.
Go to Settings > Processes, new a workflow of Product.
Then set the workflow as follow.
Check "As an on-demand process", select Scope to "Organization".
The step is "Change Status", change Product to Active.

Activate the workflow.
This way, in the product view, select multiple records and select Flows > the workflow.
