Macerate the strawberries: In a medium bowl, toss halved strawberries with 1/4 cup of the sugar. Let sit at room temperature for 15 minutes, stirring once halfway. They will release a deep red liquid and become glossy and soft.
Make the lemon simple syrup: While the berries macerate, combine remaining 1/4 cup sugar, lemon zest, and water in a small saucepan. Bring to a simmer over medium heat, stirring until sugar dissolves. Remove from heat and let steep for 5 minutes.
Blend everything: Add macerated strawberries (with syrup), lemon simple syrup (zest and all), fresh lemon juice, and pinch of salt to a blender. Blend on high until completely smooth, about 30 seconds. Taste and adjust sweetness or tartness if needed.
Strain if desired: For a seedless texture, pour mixture through a fine-mesh strainer into a liquid measuring cup, pressing on solids. Skip for a more rustic texture.
Fill the molds: Pour mixture evenly into popsicle molds or paper cups, leaving 1/4 inch at top for expansion. Fill paper cups 3/4 full.
First freeze, then sticks: Freeze for 1 hour until thick enough to hold sticks upright. Insert popsicle sticks, then freeze at least 5 more hours (6 hours total) for creamiest texture.