I think it would look unusual.

Yes it would look odd. When you start that wood feature you have to carry it up the stairs or down a hallway.

Do you have any ideas what you would put there to warm the space up?

You don't show any pictures other than part of your sofa so I have no idea of what your decor looks like but here's an idea with a small narrow demi-lune console table.

You really don't need anythng there--the space looks very narrow, and anything you put there is going to be in the walkway. And if the gate is for small children who are toddling around, a little table with tchotchkes isn't very practical.

Agree with above - if the room needs some sprucing up, I would focus elsewhere

I would not clutter up that entryway area, except perhaps a couple small, framed art pieces hung vertically on the left of the closet door, and a coordinating entry rug. Guess I'm in the "less is more" group but I sense your need to have every spaced filled with something. Let that entryway breathe :)

You would need to carry it around the corner and up the stairs from what I can see to make it fit in. What is that door to? A closet or a garage or a bathroom?

All or nothing IMO you do not just plunk trim with not thought about the rest of the spaces around it.

Go stand back, post pictures all angles of the room, please. Bet me your focus is the wrongly placed focus? : )

To answer your question: yes, it would look odd.. You didn't really post your Design Dilemma to get the best advice. The pros on Houzz don't just look at one element. They need to see the big picture.

Here is how to post a Design Dilemma to get the best results. You need to show the whole space—all four sides of an entire space—even if it’s about one specific Dilemma. A layout fix is often a Design Dilemma fix even if it doesn't seem like that. Commenters need to see doorways, windows, closets, finishes throughtout the space to make informed suggestions.

State right at the beginning what the room will be used for and who will use it. State the non-negotiables at the beginning, what has to be kept. Give some idea of the budget. And provide approximate measurements

https://www.houzz.com/discussions/6417709/unofficial-design-dilemma-boot-camp

Keep in mind that you're inviting some design pros and people with a design eye into your home via photos. If they walked into your actual space, they would take in all of it: all four sides, doorways, floors, windows, ceiling, built-in fixtures, lighting, trim--the "givens"--then the features that can be changed--furnishings, paint, art, rugs, decor items, etc. It's all one first impression. Same thing with photos. That's why it's important to provide well lit photos of the entire space, taken from all sides of a room regardless of the size of your problem.

Usually wainscot occurs throughout an entire room. Having just a small section will look odd.

It would look weird and doesn't really go with your more modern-style sectional.