Actually 2700 meters (Monte Cinto is officially 2706m) is closer to 8900 feet.
More important, Corsica is not part of Italy! The native language (spoken by 60% of inhabitants - the official language is French) is similar to Italian, but then so are Catalan, Romanche, and Castilian. Corsica has been part of France since 1768 [in theory, that's when Genoa ceded it; in practice French occupation required a war that lasted into 1769, since Corsica had been effectively independent from Genoa since 1735 if not 1730]. Napoleon Bonaparte (orig. "Buonaparte") was born there in 1769.
After the fall of the Roman Empire, the island became Vandal territory, then Byzantine, then Lombard. Charlemagne, in his capacity as King of the Lombards, ceded it to the Papacy in 775(?), who gave it to the Bishop of Pisa in 1077. However, a division of the bishopric in 1133 began a long campaign which eventually resulted in the Genoans supplanting the Pisans by 1284.
(One of many legends about Cristopher Columbus says he was born in Corsica, during the period when the island was claimed by Genoa.)
I'm skipping over Sampiero Corso's invasion/revolt of 1553 and his attempt to repeat it in 1564, and several French "interventions" in the 18th century...
Corsica today is considered part of the French "Metropole", not even an "overseas region". Its two "departements" constitute a "territorial collectivity" with marginally more autonomy than other French "regions", but for all intents and purposes Corsica functions as part of mainland France. Recent years have seen gestures in favor of Corsican language (for example, bilingual road signs), but you see those in mainland regions like Brittany too. Quite a few Corsicans today aren't happy with French rule (you see lots of those road signs with the French names erased, and there have been some bombings over the years), but they seek independence or autonomy, not unification with Italy.
The GR20 trail runs across much of the island, a friend of mine did two sections of it in separate multi-day trips, and reported several 1000m days. His GPS tracks don't show any 5000ft+ single-peak routes but you could easily get that much gain by doing multiple peaks in a day. And judging from his pictures, it's a spectacular place for hiking.