How to hide a dog door?

Yes, not the best placement, but I am sure the prior owner had a reason. My vote is to remove it. Then add some sort of embellishment that is repeated so that it looks like it is intentional and not an afterthought. Your repainting colors sound very pretty!

Do you want to keep the doggie door usable or not use it anymore? If you don't want it. It looks like your door may be multipanels. If so is the dog panel in 1 panel or crossover- hard to tell on low resolution photo. If it's just in 1 panel you could remove the dog door, cut a piece of fiber broad the size of the eased panel and glue it over the raised panel and hole using liquid nails. Do the inside also.

If it goes over multiple panels, then the hardware store has pieces of thin sheet metal with patterns cut out of them they come in silver color, brass etc. Do the same thing, remove dog door, cut a piece of fiberboard to fit bottom third of door glue with liquid nails. Paint the fiberboard and put the silver metal sheet with cut outs over that. That way the dark blue will peak out and just consider it a decorative piece.

If you do paint the front door a dark color, it should blend in much better with the smoky color of the flap, so it will not be as noticeable. We have our dog door in our back door, but it is much more noticeable because it is an extra large to accommodate the large size of two of our dogs (our smaller ones can even make it through, they are 10 lbs. and just push their way through if the larger dogs don't hold the flaps open for them).

Remove the focus from the doggie door - add a large potted plant and/or a bench seat along the side wall in front of the door. A comfortable lawn chair and two hanging baskets or wind chimes on the other side of the porch will tie it all together and create a welcoming entry. Enjoy!

they make decorative plates in stainless, aluminum, & brass to cover the bottom of door, here is one 10 inches tall https://www.homedepot.com/p/Prime-Line-10-in-x-34-in-Satin-Aluminum-Door-Kick-Plate-J-4620/202634814

I think if you aren't wanting to replace it then try painting the house and door like you mentioned. But another idea would be to move the porch rails over to the left and grow something in front of it. Then anyone walking up to the porch enters on the right side.

Ok, why don't you glue in new wood, fill any gaps with wood filler and sand, prime and paint it along with the rest of the house you are doing? Makes no sense to me why anyone wouldn't think of just putting wood back in it.