You could try concrete crack filler, but concrete is porous. I would recommend a concrete curb that would help to push the water off to the edges. Water will take the path of least resistance. You just want that to not be into your shed.

You could spray water on it with a gardenhose to see where it's leaking. If it's really coming from under the frame there's this stuff called tixophalte caulk, this is meant to be watertight.

When I used to install these we would put down a thick silicone or construction adhesive base then set the door on top of it.

What does the concrete slab look like? It's not running back towards the house is it?

It seems like the water is probably getting higher than the sealant. At that point, it's almost certainly a manufacturing thing with the door that you can't easily fix. A sill would be the best bet imo.

Without a real sill and a slope on the outside your going to have water intrusion. Add a gutter/diverter above the door on the roof. Remove all the sealer you can. Use waterproof concrete sealer/paint on the slab. Fill every gap you can with silicone. Shave down the slab with an angle grinder to add some kind of slope.

If the concrete applier 'forgot' to add a slope to the outside see if they will help with the remedy.

Do you have eave troughs there by chance?