but how can I add a field in the product sheet that is shared with its variants?

product template represents a blueprint of the product as long as you don't have any variant of the template
product variant and product template are the same but when you create product variants of the template it's don't same unique attributes like barcode, internal ref, weight, volume 

each product variant can have a unique internal ref, barcode, etc...

that's the standard behavior you could not change it but need something that shares among the variants define a new field on the product template to achieve it