With more Google research, several places talk about removing all the finish. The finish is still patent where the non grubby hands did not effect it. Would I need to use a stripper to get the rest off or just re coat over it? With what?

The wood on my chair was dirty. No stripper needed. I've refurbished "one" ladderback chair with a rush seat. I'd lightly sand, a quick once over with fine grit sandpaper...then spray lacquer. Back when I did this...I got a book from the library on "how-to-weave a rush seat". Now they have a Utube video Link One roll of rush is enough, 210'. It turned out really nice! Link to Rush, use a coupon to get a discount

Yes, I have watched the YouTubes and ordered rush from Amazon as that was cheaper than a minimum order from JoAnns. I just wiped some stain over the bare areas and will probably hit with some poly later on.

I saw Lone Fox revive an old coffee table using this product on a youtube video. The result was remarkable.

Just clean it. Leave the areas where “little grubby paws” wore off the finish. I’m not a fan of fake worn finishes, but finishes actually worn by time and by people you know and love are charming and heart-warming to me.

I purchased tie product @ffpalms posted. It worked well on all of my antique furniture. Did not do my children's rockers though. I wouldn't put any poly on it, if it were mine.

Looks to me the original finish may have been clear lacquer which is different than polyurethane.

You could spray it.

https://youtu.be/oKwEp7C3R2s?t=20

I have the Howard's Feed N Wax and it's a good product. It will help condition and give a wax finish to the worn portion, but it will wear off and what it will do to the finished portions will depend on what the finish is. You can try it, but if you put wax on it and need to do something else, you have to strip off that wax.

I've also used Howard's Restor A Finish. From what I understand, it's basically a solvent (the one I have used is tinted) that works to take existing finish and smooth it out. I've used it to get old paint and gunk off an old radio cabinet and touch up other furniture. It can me anything from a miracle to meh, depending on what you are working on.

Another product I really like if Odie's Oil. I used it on three projects last year. I did do some light sanding, but I didn't take of all the finish on any of them. Each one had wear that required a bit of attention -- splintering, deep scratches and the impression of embossed vinyl. Odie's isn't cheap, but iit goes a long way and I was really impressed with the results. This is the top of an old sewing machine cabinet that butterflies open. The foreground is with one light coat applied and the back is just lightly sanded (I don't know what the finish was, perhaps an old shellac, but the sanding made it look chalky). I had no idea that wood grain was there -- it just looked average old yellowy worn and scratched wood. Then Wowzers!

The Odie's brough out gorgeous wood grain in what would have been 3 throw away pieces. This cabinet and the working black Singer sewing machine inside came from a thrift shop for $25. The otthe cabinet had been stored in a Texas garage with an embossed vinyl seat stool upside down on it for years before I got it and the other was my mother's former office desk. My XBIL sold off the furniture from my dad's former office cheap -- and no one bought it. I couldn't stand to let it go when the property was sold, so I brought it home and made it my sewing table. I couldn't believe how great it looked after he couldn't give it away. My win! LOL I'll try to find a picture of each of those, but I was so pleased that I was considering working on my grandmother's mahogany dining room table with it this year. One thing to know -- the Odie's bonds with the wood --it doesn't just sit on top, so if it's something you will want to change the look of every few years, I wouldn't go that route. I'll try to post some other photos in the morning.