Ponding on doorstep-grind surface/slope? Fill +paint?

[Safety First]

The sealant under our threshold strip is failing to hold ponding water on the surface of the blue lias doorstep.
Overall the doorstep is level/sloping away areas. None actually sloping back towards the house, it does have some small pits- maybe enough to hold a 50p piece...

What would you do?

Fill it with something and paint it (black bituminous?)

Grind a slope in it?

Any advice?

Cheers all,

 

[John D v2.0]

I'd clean out the sealant and apply some new.
It's not ideal to have no fall but the sealant is designed to prevent water entry.
Are you sure that's where it's coming through, sometimes it can be getting through inside the frame if the drain holes are blocked.

 

[Safety First]

I'd clean out the sealant and apply some new.

Just have done..
I just think I am asking rather a lot of it...

Are you sure that's where it's coming through, sometimes it can be getting through inside the frame if the drain holes are blocked.

Poured water through the drain holes- not blocked...

Guess I'll wait and see...

But maybe someone has filled and painted their doorstep?...

 

[John D v2.0]

But maybe someone has filled and painted their doorstep?...

I'm not sure that would work as then you'd be relying on the thin layer of filler and paint, but as long as it can dry out quickly it should be ok.
Hope it's ok, see how it goes.
The ideal solution is to install a canopy to keep the worst of the weather away from the openings, but that's a big step