Using Pages to make Business Cards, how to quickly copy first completed template to all spaces?

I would start with a blank Avery business card template so you don't need to deal with the prefilled nonsense of Apple's Pages Business card templates.

https://www.avery.com/templates/category/business-cards

Choose the blank template that best fits your needs and click its WePrint link which will reveal the downloadable blank Pages template for that particular Avery business card.

To get a single card formatted may take several text boxes overlaid and grouped. Text in each text box may be aligned differently for layout purposes dictated by your card design. Each selected text box must have its Text Wrap set to None in the Arrange panel, so they can be stacked over or under one another. By default, text boxes have a transparent background.

Once you have a single card constructed (grouped), then you can press the option key and drag that card to its location in the business card template. I recommend that you position two cards adjacent in the first row, and then group each selected card in that row. Then you can use the option key to drag an entire row of cards downward to fill out the rest of the sheet.

If you plan to have these cards professionally printed on 15 pound index card stock (so they don't flap like a fly's wings when handled), then consult with the printer on what they expect for the source document. It will be PDF and Apple's Pages may not be able to produce a proper PDF to meet the printers input requirements.