Do gemstones such as emeralds, amethysts etc have any purpose in Skyrim other than being sold for gold?
Gems are useful for smithing, as they increase the skill by quite a bit. They can be smithed into jewelry which can then be enchanted with properties that buff certain attributes (fortify light armor, extra sneaking or lockpicking percentage, etc).
Also, some gems are used during quests. For example, there is an Argonian in Riften who will ask you to find three Flawless Amethysts so that he can make a wedding ring for his lover.
As you mentioned, they can also be sold for a fair amount of gold. Keep in mind though that gemmed jewelry is more costly than just gems alone. If it’s money you’re after crafting jewelry and selling that would be more advantageous than selling the gems alone.