I would try a ground cover like Lamium, somewhat open and very easy to control. If you want something taller, try a hosta or pulmonaria.

Is there any kind of shrub that would grow to fill the "hole" in the arbor vitae? Not just a ground plant, something to fill that gap between the tree and the porch and fill in the dead spot?

I suppose if you cut off those lower branches you might get a small boxwood in there, though the competition with the arborvitae's roots might be a problem. Not what I'd do. And it's not at all clear to me what kind of privacy that narrow plant could possibly provide--and just to a porch entry.

I see from your 2nd picture why you'd like to keep the arborvitae -- pretty (the 1st pic makes it look like leaning a bit...?) Possibly you might shear off the bottom ratty stuff, place a trellis [white or grey to match siding or stair railing -- something to match the latticework of your staircase would be lovely] diagonally behind, and plant a young climbing hydrangea or other tidy-ish shade-tolerant vine to fill/ balance the space attractively. ('young' so that root ball is small enough for you to fit it in w/o disrupting arborvitae roots too much :)

A bushy evergreen shrub about 3 ft high is what you want. Like the ilex (?) at the other end perhaps?