Is there a public database for annual flights by aircraft type?
Short answer is "no". There is no such thing as a "registered" flight.

In cases where pilots have filed flight plans, you could aggregate that data. It would be a BIG programming project. There are commercial services that analyze flight plan data. They might be willing to help you for a price that would be in the 10's of thousands of dollars.

Flight plan data would only be good for US information. Other countries would require other data collection.

Small planes usually don't file flight plans, but you may not care about this.

Aircraft manufacturers estimate how much their planes fly around. Whether they would be willing to share this data with you, I don't know.