Replace the glass with wood that will be even with the existing wood.

I agree with suero. At first I considered suggesting dropping tiles in there (perhaps openings are 12 x 12 with a lip for tile to sit on?), but tiles may be a bit too casual for the look you're going for.

I found a coffee table like yours that someone refinished. It will give you an idea what the legs would like painted. The top isn't what you're going for but they do mention the problem they had with plywood inserts warping.

Does the table have sentimental value for you? It might be a lot easier to find an oak table on Craig's List that's more in keeping with the style of your chairs. Those table legs speak French.

Oops! I always forget the link.

https://nevernewthriftydesigns.wordpress.com/

Scroll past the chairs.

Why don't you take the table into a glass shop and have them cut a top that is one piece to sit on top of the entire table(leaving the supports)? I took an end table that was a hexagon that was missing the glass and they cut one that was perfect!

Since you don't want the glass and it looks broken, have you thought about going ahead and taking all the glass out, but leave the cross pieces for now. Then post a close up picture of how the edges that support the glass look. That will help you get some more ideas.

You could add stainless steal or copper panels cut to fit the squares. You could use wood planks and add a herringbone pattern inside the squares and gel stain the whole table to match or add mosaic tiles in squares.

You could repurpose the slate from an old unwanted pool table. It's a lot of work but some are even recycling old pool table slate into kitchen countertops.

Ditto, Linelle. I hope she got this table for free, because what's she's considering is going to be alot of work. The easiest way to fix this table is to remove the glass and cut out the grid. Install a ledge underneath the table and cut a piece of plywood to fit; furniture grade if staining; paint grade otherwise.

She said that the table belonged to her mother and she wants to keep it.

Stained beadboard. It would first need to be glued onto a piece of plywood which fits into the opening with vertical and horizontal supports. I can envision it but can't specifically explain how it can be done. ;)

You could make a wooden 'slipcover' to go over the entire top

You could use furniture-grade plywood, add apron &

stain

I've also seen where they used laminate flooring on non-furniture grade plywood as a tabletop

Here's another example

http://www.rachelteodoro.com/2015/06/easy-diy-planked-table-top-cover-for.html?m=1

Is thee some way the inserts could be tiled with some beautiful tiles?

Slate or some other stone tile might look good in those cutouts.