Quarter 6 ounces (about 1 cup) strawberries, then slice quarters crosswise into very thin pieces. In a mixing bowl, combine strawberries with 1/2 cup sugar and alcohol and let stand in refrigerator for at least 2 hours and up to 2 days.
In a countertop blender, process remaining strawberries until very smooth, about 30 seconds. Measure and reserve 1 1/2 cups purée. Extra purée, if there is any, can be put to another use.
In a clean mixing bowl, whisk together 1 1/2 cups strawberry purée with heavy whipping cream, corn syrup, and remaining 3/4 cup sugar until fully combined. Add salt to taste.
Chill in refrigerator or ice bath until base is very cold, at least 45°F (7°C), then churn in ice cream maker according to manufacturer's instructions. In the last minute of churning, retrieve strawberry mix-ins from the refrigerator, strain off syrup, and add mix-ins to the churn; reserve strawberry syrup for another use (it makes a great coffee syrup). Transfer ice cream to airtight container and chill in freezer for at least 4 hours before serving.