With the availability of affordable 4k screens, I think there's a real opportunity to get a service like this for cheap. I think high resolution (and a matte screen) are key for appreciating art. A square ratio might be ideal to display any sort of painting, but 16x9 is by far the most available and least expensive, especially for high resolution displays. depending on room size and mounting location, I think a 24 or 27 inch 4k monitor could be perfect, since you get really good pixel density. Oled could be interesting for this use case, though they are mostly larger TV displays.
I download a lot of art as desktop wallpapers on my 4k desktop, and then use photoshop to get the correct aspect ratio. With many art pieces, especially with a lot of dark backgrounds and centered subjects, content aware fill can be used to create a 16x9 ratio without cropping any of the content. For other art, like movie posters, simply creating a [black]/[white]/[color picker] color background for the poster to lie within allows for 16x9 landscape ratio from an originally portrait composition.
I usually use advanced google image search to set my parameters for ~12 megapixels (4k at 3840x2160 is ~8.8 megapixels, though extra size can be beneficial if you are going to crop, or for extra detail for content aware fill), though it isn't perfect and often displays images that are not the target size. it still helps cut through some of the tiny images that meet search parameters. I've found that many of my images are just from Wikipedia, I wonder if Wikipedia's API could be leveraged to retrieve a set of paintings from the rococo period, "renaissance art", or whatever. Using Wikipedia images might help avoid some legal issues, though i'm not sure if the API allows for commercial use.
here is the original of "David with the Head of Goliath", by Caravaggio (This version is a bit smaller than the one I originally downloaded, I'm not sure where I first downloaded it)
https://upload.wikimedia.org/wikipedia/commons/7/70/Caravagg...
Here is a version that I used Content Aware fill in Photoshop to widen the image, notice that some of the patterns in the cracks of the painting are repeated, especially on the right. The output isn't perfect, but its organic enough to my eye, and avoids any black bars.
https://i.imgur.com/pbr3wXZ.jpg