Do I have to share my track logs and score my flight to buy insurance through Starr Gate?

No.

Renter pilots can buy insurance through the app, without sharing and tracklogs or scoring any of their flights. Scoring flights can help pilots earn discounts, but it's not mandatory. The app makes buying insurance easy, even if pilots choose not to upload any flight data to the app.

Also note, pilots who do score their flights will never see their rates go up.