Search on here and also google about painting tile -- it can be done.
You really do like the minimal to no color look! In your inspiration pictures, some rooms are warm and some cool. I think that your flooring calls for a warmer tone - light warm gray, or a rich off white as the first of your inspiration pics show. Don't match the trims in the rooms, go with a deeper toned white.
If you want to try a rug, look for indoor-outdoor if you are worried about spills. Many have nice texture for indoor use. Also I have found that a good quality wool flat weave is pretty easy to clean -- if you notice the spills pretty quickly.
I agree with more lighting in the kitchen -- a bigger or multiple pendants over the island (maybe a slightly smaller version of the dining table chandy?) and since the cans don't seem to be optimally placed in the kitchen, do see if you can add undercabinet lights. I recommend 3000-3200K LEDs which produce a nice white that doesn't lean blue like "daylight" (5000k) bulbs do.
Q