Answer: To be honest I am surprised your attempts have not succeeded, although I suppose it is possible that hosta seedlings have sprung up here and there or you had a particularly sturdy variety of groundcover (slightly stoloniferous) hosta. Assuming it is indeed hosta, I think I would try again with glyphosate; be very careful to read and follow the application instructions exactly. The chemical must be allowed time to be absorbed by the leaves and then translocated to the roots. This can take two weeks and looks terrible while you wait, but it can't be rushed. Since it needs to be absorbed by the leaves, do not cut the plants back or otherwise impede their growth prior to application, either.
After the chemical treatment and full waiting time are over, I would cover the area with a layer of newspaper or an air and water permeable weed barrier fabric topped by an inch or two of mulch. Since dogwood trees are shallow rooted and strongly resent any root disturbance (including excessive mulching), I would be cautious about digging or otherwise disturbing the area beneath them. The hostas probably served as a nice insulating layer for the tree roots, helping to keep them shady and cool, so you might consider maintaining a light mulch layer there to serve the same purpose.
If there is any doubt in your mind as to whether or not this is actually hosta, you might wish to take a sample to your County Extension office for a positive identification. In case that's not what it is, they may have some additional control methods to suggest as well. Their telephone number is 784-1001.
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