I believe you have to burn it anyway to do the carbon dating.
Yup!!!! One of the "old ways" to do C14 dating was to use liquid scintillation counting. The first step WAS to convert all the carbon the CO2, then react it with lithium metal to form lithium carbide. Add water to convert the carbide to acetylene, catalytically convert the acetylene to benzene, dissolve the benzene in the liquid scintillation "cocktail" and count the radioactive decay. Complicated, no??
Today, they use high-resolution mass spec directly on the CO2 and measure ALL of the C14 and C12 without the need to wait for the C14 to decay. MUCH more sensitive, accurate, and less prone to error.
BUT, you are correct, in that both methods required "burning" the sample to get the carbon into the form of CO2 first.