To help save money, you may want to simply look at updating the wall color (even to a bright white) and bringing in some modern bathroom accessories and small artwork to distract from the countertop

I would paint the wall blue or green in a matte finish. A large colorful print over the towel rack would look good.

I would not add in a frame less door if you are not replacing the tile. It will most likely get ripped out by the next owner.

Freshen the grout with a good cleaning, add some hardware to the vanity, freshen the wall paint and add a pop of color to the rug and accent towels.

I never say white paint, but I think it will look nice and clean.