Need a suggestion to fill the space that was routed out on the doors.

What is the space shaped like? Can you post pics?

Are you wanting to decorate/update just that center part of the doors? If so, you could paint the insides and use stenciling, or just stencil inside. You could use adhesive wall paper or contact paper to give it a design.

https://www.bhg.com/kitchen/cabinets/makeovers/easy-kitchen-cabinet-updates-from-kitchen-makeovers-magazine/

Are you planning on painting the doors? If you aren't, the routed areas would show not matter what if you want to leave it the natural wood. If you are planning on painting them, clean the routed areas well, I would sand it lightly and fill it with wood filler. Make sure you sand it smooth and feather it out so that the lines won't show.

Use wood putty (filler), let dry and follow instructions for painting.

you can use a wood filler and sand smooth, but it will affect the appearance of the surface, unless you plan to paint the entire area.

Talked to hubby about your problem, he came up with something that might work for you. He said to get a piece of hardwood very slightly smaller than the width of the area, a couple of inches long, high enough that you can hold it. Wrap it with sandpaper, he said start with 220 grit and then 440 grit to finish. It will be labor intensive because you will be working only a small area at a time, but may work. If you make a few of the hardwood pieces you could enlist help with the job to try to get it done a bit quicker. I asked him about using any of the small electric tools that are around and he said that they would only sand the middle because most of the sanding bits you can get are round. Sorry about that part, no easy way out. I hope that this will let you clean out the routed areas of those horrible lines. The wood is so beautiful, I don't know why someone would cut through wood grain like that!

You could try wood putty but I find it difficult to work with. It is dry and crumbly out of the can (?brand) but dries hard as steel. It's almost impossible to sand after it's dried and the front of your doors won't hold up to vigorous sanding. Perhaps if you could find a way to thin it slightly it would be easier to work with. Drywall patch would fill the area but may not be durable enough. I'm guessing you're getting ready to paint....wonder if mixing a dab of paint with drywall patch would add strength? I'm sorry if these suggestions sound off the wall but I experiment until I find a solution. Have refinished furniture for years and did my own home maintenance too. You could have them refaced but I'm not sure about durability or cost. If you solve this problem I highly recommend buying a paint sprayer or at least use Floetrol to minimize brush marks. Rollers won't give a smooth finish but some sites recommend a foam roller, or a roller them brush over that immediately. Good luck!

Idea 1: If you found some trim that you liked the look of you could fill the routed grooves with wood filler, drywall goop, glued pieces of wood or plastic or even heavy cardboard, then apply the trim after you re-stained the "good" wood. The hardest part would be following the bent lines at each corner.

Idea 2: I would try to find paints the 2 colors of the good wood and paint the grooves with tiny paintbrushes like you use for painting pictures.

Use the lighter color for the whole groove and add the darker color to mimic the patterns of the wood. My aim would be to try to make it seem as if the entire thickness of the wood had the same grain as the surface, instead of having the dark rough wood underneath.

Judging by how I have fixed up my own kitchen, though, what I really would do is pretend I don't see the grooves.

Good luck!