I noticed that we have a decent batch of opportunities that have a listed Primary Contact, but the Account field is not populated, which means that the Opportunity is not linked to a household.

the default NPSP behaviour is to copy the Account's Primary Contact (from the Primary Contact field) onto the Opportunity, but not the other way around which is what it sounds like you are describing.

You can update the Account records in bulk using any data import tool, such as the Data Import Wizard or Data Loader.