Quarter the potatoes and place them in a large stockpot, then cover with cold water by 1 inch and bring to a boil over high heat. Once the water boils, add 1 tablespoon salt and cook for 13-15 minutes, until a fork or paring knife slides in with gentle resistance and no firm center.
Whisk the mayonnaise, sweet pickle relish and its juices, mustard, apple cider vinegar, celery seeds, paprika, 1 teaspoon salt, and pepper in a medium bowl. The dressing should look smooth and glossy, with the relish evenly distributed instead of sitting in watery pockets.
Chop the hard boiled eggs, celery, sweet onion, and dill while the potatoes cook. Keep the celery and onion in small, even pieces so they bring crunch without overpowering the tender potato texture.
Drain the potatoes completely, remove any loose peels, and cut them into 1/2-inch chunks in a large mixing bowl. Allow excess steam to escape before dressing them; the potatoes should be soft and lightly crumbly, not dripping with water.
Fold the dressing through the potatoes with a broad spoon or spatula, then gently stir in the eggs, celery, onion, and dill. Stop once the potato chunks wear a smooth, creamy coating, then taste and adjust with salt and pepper as needed.
Finish with fresh dill and paprika, cover the bowl, and refrigerate for at least 4 hours. After chilling, the dressing will look slightly deeper in color and the mustard, relish, dill, and celery seed will smell more unified.