I thought this would be something reasonably natural to say in Spanish
Indeed it is! You are completely right in your assumption.
I'd say we tend to use an article when refering to more than one family member (e.g. ayer fui a visitar a los abuelos), and a possessive otherwise (e.g. ayer fui a visitar a mi abuela). Of course there would no problem in saying, ayer fui a visitar a mis abuelos, nor, ayer fui a visitar a la abuela. Unless there is complementary info (e.g. ayer fui a visitar a la abuela de mi novia) it is always understood that one talks about one's own family members.
We also use an article with body parts, for example, me lastimé la mano, me raspé la rodilla, me duele el estómago.
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