In most respects, with the exception of the maker, your grandfather was correct, but so are you.
It's all about terminology. Up until the the start of the 20th century, and perhaps a lot later, the term 'plate' referred to solid silver. The term pre-dates the invention of silver plating and was an old throw back to the days when your silver was often made up with that what was connected to the dining table. It gradually fell into disuse because of the confusion between 'Plate' and 'Plated'.
Your spoons are made of sterling standard silver, assayed at Exeter in 1831. The maker was William Sweet.
It sounds as if your grandfather had a wonderful collection, I hope you'll show us a lot more of it. Remember, it's not necessary to have a question about something to post it on the Forum, if you anything that's slightly out of the unusual, we're always pleased to see it (but please try and keep your images to a max. of 7 inches, for the benefit of those with smaller monitor screens).