Some inspos for you.

Please stand with your back to the window with the lace curtain and take a photo of the adjoining space. Also as Lisa asked: What is the function of the room? I like her images that show a mix of mostly white paint but with some wood features. That might work well for you,, using the back entryway as a transition zone with a mix of wood and white paint. tNote the wood shelf below. You could also have a wood oak top storage bench built under that window.

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The main function of this space is as the main entry to the home from the garage. The closet is used to lots of storage for the kitchen + cleaning materials and backup pantry items, dog food, etc. I added 3 more pics to the orig post that show the adjoining space. I'm mostly concerned with ideas for the closet doors/trim and matching wainscoting - paint only one or both or neither? I do need to replace the flooring as it's curling up due to years of moisture in winter.

What do you think about removing the entire closet and doing a built-in bench/cubbies/upper cupboards in that area?

Paint it all and since you use the closet, I would put the bench and cubbies on the pony wall where the hooks are.

Is that electric baseboard heater used?

unless you did the closet like this but that would be extra expense.

The heat in that room is used and it hot water.

Lisa and Housegal posted some beautiful inspiration pics up there. I hope they help you. I personally like the look of wood finishes in the right context and in the right home, but if it really bothers you, you can always paint over it. That seems like a lot of extra work though. If you can make the wood work for you, you may grow to like it over time perhaps. Here's an article that helps explain which paint colors work best for that honey oak colored wood. Good luck!

https://www.kylieminteriors.ca/the-best-paint-colours-to-go-with-oak-wood/

I had imagined far more oak features, but you have plenty of painted white, so just paint most of the space white, leaving some oak features.

Keep the doors as a closet, not open cubbies since the space opens onto your dining room and living room, so it'll look tidier. Maybe set aside one closet space with just hooks so the family doesn't have to deal with hangers and will find it easy to hang coats in the closet as they come in.

A white storage bench with oak seat under that window would be a great place for stuff--offseason shoes, etc. Or just a straight across oak bench with baskets below for coralling shoes. If you keep the closet doors, the window bench and closed closets space can be a nice extension of the dining and living room areas when you have company.

I would paint all woodwork white and paint walls the same as kitchen. Do away with the hooks where backpacks are. Place a bench over the baseboard heater. Store shoes under the bench.
I like your glasses on display, I have some of them.

I love the wainscoting. Can you stain it a deeper color?
Paint everything else white.
Remove hooks and use the above inspiration to create a mud room effect.