How to fill gap in front door trim
The metal sill was sized/cut incorrectly.  It should have horns, cut at where the stop trim is, extending under the jamb, to the full rough opening of the doorway (to under where the jamb trim piece is).

Also, the piece of wood the leading edge of the metal sill is resting upon should not be sitting so proud.  If this piece of wood can be recessed, so that only about 1/2"-3/4" extends forward of the metal sill, this may solve (though not ideally) the issue of the gap.

I'm concerned there may be a hollow space along the entire sill that isn't particularly well sealed from water infiltration.  You may want to remove the aforementioned piece of wood and consider what is required to seal this area with a 1x or 2x flooring/sub-sill/framing.