This is going to get complicated, but that's the nature of A Song of Ice and Fire. It goes without saying that this will contain spoilers, I will do my best to mask them appropriately.
Robert did not hate Targaryens
Not in the classic sense of "they must all die." His rebellion was precipitated by the actions of two specific Targaryens, namely Rhaegar and his Father Aerys II.
Specifically, Rhaegar ran off with Lyanna Stark, sister of Ned and Brandon Stark and wife-to-be of Robert. Brandon went to Kings Landing to get his sister back and was taken prisoner by Aerys. When Rickard Stark was summoned to Kings Landing to ransom his son, both father and son were brutally murdered by the king. At that time, Aerys demanded that Robert and Ned be handed over and Jon Arryn of the Vale, who was fostering the men at the time, raised his banners in revolt.
What does that have to do with Robert not hating Targaryans?
In A Clash of Kings and A World of Ice and Fire, it is stated that Robert's Grandmother was Rhaelle Targaryen (this is important and I'm slowly getting to the point).
Why does that Matter and weren't we talking about money?
I'm getting there. It matters because, while anyone (Jaime Lannister, Ned Stark etc.) COULD have taken the Iron Throne, any claim would have to be legitimized. Robert's claim was easy to legitimize because
Rhaelle was the daughter of Aegon V which made Robert a direct descendant of a Targaryen king and gave him a blood claim to the Iron Throne-as long as no other children of Aerys II lived.
So his claim was only valid as long as he was the most eligible member of the family. Viserys and Daenerys both had a stronger birth claim which is why Robert wanted them dead.
For the love of all that is holy, I asked about coins!
I told you I was getting there. Robert kept the currency of the Targaryen dynasty intact because he was sort of a member of it. He was not implementing a new system, he was continuing the old one under new management.
This was important because many of the Houses of Westeros supported Aerys and his children. Replacing every Dragon with a Stag would be pouring salt into a fresh wound when he needed the support of the Great Houses to cement his rule.
Furthermore, the only banks mentioned in the series (book or TV show) are in Essos. The Westerosi seem to like to hold on to their cash which means replacing coins would require someone to go house to house, village to village, kingdom to kingdom from Dorne to The Wall and swap out EVERY gold coin in seven kingdoms.
At the end of the day, gold is gold and it would have been politically and logistically difficult to change out Dragons for Stags.