Try "recessed finger pull" and "cutout cabinet pull" In the meantime, you can swap one of the less used four with the busted one.

This may be helpful https://www.remodelista.com/posts/remodeling-101-cutout-cabinet-pulls/

You might be able to substitute Grommets for this. Look at Doug Mockett

https://www.knobs4less.com/Cabinet+Hardware/Recessed+Pulls/_/N-category+Recessed_Pulls+shape+Round

Thanks for all the ideas!

I wonder if a wood veneer strip would work inside that hole, if it’s not too stiff. That would be preferable to painting. Actually, I think the door is solid wood so maybe it just could use a finish coat of poly or wax or something.

The hollow grommet part of the pull is still intact. I may try using glue dots to keep it together. Caulk would work too but would be more permanent.

The pulls that I have now block the view of the inside, which is what I want.

I may also consider making some kind of pull out of wood with a router and hole saw later when I have more skills. Or some other material if I ever get a 3D printer. The hole size is an issue for the off-the-shelf options I have found so far, and I don’t want to make the hole bigger. These are nice cabinets that have held up for years and get used a lot.

The issue with leaving these holes wood, which I have seen in some handsome mid century kitchens, is that the wood around the hole will darken from the oils on your skin.