My wooden screen insert has two crossbars about 1/3-1/2 the way up...I've also seen metal grilles you can buy from some places made of various metals.

Vintage advertising push sign that spans the glass?

Here is a link that might be useful: Holsum door push

I have seen beautiful metalwork screens that span the middle of the door - they are usually heavy bands of brass or bronze and are "woven" in a diamond or square pattern with small loops top and bottom, They are sturdy but open like screens - there is an antique shop in Louisiville, Joe Ley's, that has them in the damp, scary, filthy, cob-web infested basement - wear a filter mask if you go, theres crud everywhere...