I like that you painted it white!

I think Ikea cabs with lots of lower drawers are MUCH better, storage wise, than builder grade cabs. I have built-in-place builder grade with fixed shelves, and lots of wasted space from huge filler areas in the face frame. I only have 2 normal size drawers, and 5 little ones that are, literally, 3.5" wide inside. The drawer glides are low end, and not super smooth. Interiors are beat-up low-grade melamine particleboard, not looking too good. But if I were moving in 5 years, I would only paint (the doors AND the interiors that are beat up). Because, when I organized my kitchen, getting rid of everything I don't use, and adding a few organizer shelves from Container Store, it became very easy to find & access what I do use, even in my tiny low-end kitchen. If I were to retrofit 2 pullouts in the lowest shelves of 2 36"-wide cabinets, I'd be close enough, functionally, without the expense and hassle of replacing the cabinets.

Just one last clean with lacquer thinner should do it. Then I would do wax. I have found that those scrubby pads never hurt oak or the finish on them. I would never use them on fine furniture especially with veneers. It's going to be a lovely piece. I don't know why anyone would tell you it's not Arts & Crafts. It's not Stickley, not Roycroft but could be one of the smaller companies. I haven't dealt in A&C for a while, it could take some time to dig out a book or three. You could try a library that has a good selection of reprinted catalogues. I find researching antiques especially furniture online tedious & usually unfruitful.

The cabinet fronts are beautiful. Just touch them up with a stain pen if needed. If they are grody inside, you can paint the insides white. Or shellac them if they are just stinky. You don't even have to do a good job unless people will be looking inside. :) I love those bamboo handles, but they don't go at all with the kitchen. Leave the hinges (they blend in and would be a hassle to change) and go with a black, bronzed, or brass knobs/pulls. Your kitchen looks like a high-end custom kitchen, just somewhat vintage. Get it functional and spiff it up so it looks good. Let it be what it is and don't try to get it looking current because it won't.