Second vote for Straight Up Chicago podcast, specifically episode 53 regarding this topic.
I accept Section 8 tenants at my properties and can confirm that the process does take awhile. That said, I think it's worth it. The paperwork can be a bit of a pain, because if you fill in the form and have a "minor" error, CHA will reject it and you have to start the process again. Just takes diligence. Regarding the inspection process, this is what takes the longest. In my experience it usually takes 1-2 weeks to get an inspection after the paperwork has been processed, then, when the inspection does occur, the inspector will almost certainly find something that needs to be addressed, and it takes another week for the re-inspection. The $1,965 rental rate you're seeing is likely the county maximum for the number of beds/baths. You will have to specify how much you're asking for rent and have that amount approved. It's possible they approve for less than the published amount, so don't bank on that.
If you do go this route, my suggestion is to continue marketing the townhome for rent to non-subsidized tenants even after you have identified a Section 8 renter and have started the process of getting them/the townhome approved. Make it clear to the Section 8 renter that this is what you're doing.