If it was my kitchen I would be looking for something in a solid colour, probably a cream. I know, harder to keep clean but I can't think of a pattern that would work without picking up on those browns that your trying to minimize.

I would look for a more muted pattern - maybe tans/creams with a bit of the red color from your outdoor chairs for the kitchen runner. Leave the floor bare under the small table, but change the chair cushions to a solid more muted version of the outdoor chairs. It will visually connect the spaces.

i would also lean towards a solid. You have coral/red on your deck chairs and also in your kitchen towel. Would that color work with the cab/countertops? if you go that route, I'd add a big square colorful picture over the side table that pulls all of the colors together.

You need some texture, and color. Sisal braided oval with color banding- same style in a runner. I would go red, but you need to duplicate what ever color you choose. The human eye likes to 'bounce' around, high to low ( or any combo.) Pick up the choice color in art, candle sticks, runners etc. Your hitchcock chairs, table base side table are leading in only one direction right now. Runner size choices are few. Assumption is you have between 36-42" in the galley? Oval size rugs are plentiful. Chose large enough to have at least the front legs of the buffet on the rug. good luck.

This is where I would say less is more and not have rugs. My mil has some fancy anti fatigue mats that almost math the kitchen wood work if you need that near the sink. To me it is just easier to clean without rugs in eating and cooking areas.

I do not like rugs in kitchens at all IMO they are a tripping hazard . As for the table area check out chilewich they do custom sizes