I have a dwarf with broken tissue and an infection, but without the "diagnosis required" flag. I have a fully equipped and staffed hospital. Is there any way to treat an infection or will I simply have to wait whether or not he survives?
I don't think you can force them to do anything, so you best bet is just to make sure your hospital is working and wait for it to (hopefully!) happen.
I've only once had a fully functioning hospital system, it was on a map with such a high injury rate that I was able to have a full 2.5 man hospital detail.
So my suggestions for a working system:
This system has worked for me in the past, but even then, dwarfs don't really seem to believe about caring for others, so I wouldn't have high hopes for his recovery.
As far as I know, there is no way to force the treatment of your dwarves. The diagnosis labour seems to be a low priority task for your chief medical dwarf. These are the steps I take to try and mitigate this:
This way your chief medical dwarf will always be close to your hospital and shouldn't be distracted by other tasks.