A great and well made shower curb does all that and is fully waterproof.
Shower door curb edge finish.
I do not recommend finishing any tile edge with caulking.
I think you should tile the whole top and front of the curb.
As you might expect different people use different methods to build shower curbs.
Glass tiles tumbled stone tiles and some porcelain tiles have edges that do not require a separate edge piece.
If you cant find a suitable corner round you could finish the edge of the tile at the front of the curb or finish the front edge corner if you choose to tile the front side of the curb which i recommend with grout.
This means it doesn t take in water through leaks allowing it to last longer.
They can be installed right up to the edge of the shower and need no transition to.