Your question here suggests that there is a problem with hosting images that are not on "placeholder" services...
I'm not sure how it was obtained and what is the background. As some
other images that show up around.
My question: Should we allow those?
So, first up, what's wrong with having images that are not on 'placeholder' systems? Copying an image from one location to a placeholder does not change the "source" of the image from an "ownership" perspective, it just changes the hosting service. In the example you give, it does not change this statement at all: It belongs to the Illinois Institute of Technology.
So, the hosting location does not affect the ownership of the image. In many ways, adding the image to a hosting service is probably worse than linking the original location, from an ownership perspective - is it OK to make copies of someone else's picture? I am not a lawyer, but I suspect not (depending on the copyright licenses available).
So, the underlying premise of your question is flawed. It makes an assumption and then asks a question based on that assumption - but the assumption has to be shown to be accurate first, before the questions can be answered.
In addition, even if your assumption is true, that images should be hosted, why do you ignore the existing relationship that Stack Exchange has with imgur? There's a blog about it: New Image Upload Support.
When you upload an image on stack exchange using the special add-image icon here:
... it gets added to the special domain
i.stack.imgur.com. Like this:
If you link an image "from the web", like from a wiki page on "image", you get:
So, if you are going to suggest hosting services, you should at least mention the one that is already implmeneted and that most people use without even knowing it.