Looking at the pics again, I agree with Michael. If you haven't had a problem with water coming under the door threshold, then it's probably sealed well enough. His fix sounds very appropriate and relatively simple.

I wouldn't be terribly concerned about the insulating value of any material for that small area. Wood or grout will all be at least as good as the adjacent brick.

I started out with nothing & I've still got most of it left
M60 VP160 QS8 EP350
M22 VP100
Algonquins