Sorry to disagree, but: Yes, I would. I often do. It hones the handling skills.
As often, I don't write an essay on each point
It depends on the flight.
If I was flying from say Goodwood to Southend, OVC008 at each end, and done mostly at 2400ft as one does in that area, I would not be worried. In fact I would probably not use the AP, because when flying "real IFR" one spends so much time in VMC that one doesn't get enough hand IMC practice. On my last big trip (Turkey LTBH via LDSB, and back via LGMT, LDSP) I logged about 30hrs I think and barely a few minutes "instrument flight", which is not unusual for high altitude IFR with an appropriate icing strategy.
I think that when flying IFR/airways one should aim for the most professional conduct, which means flying accurately and generally being on top of things w.r.t. ATC etc. This means making best use of cockpit automation. The place to practice hand flying stuff like NDB holds is in Class G, at 5000ft where few in UK GA have ever ventured, out of the way, in real cloud, and one can do them over and over without anybody else caring. I do that about once a week on little local flights. That is also the place (with the 0C level being no lower than 2000ft) to see how much one's IAS drops versus the thickness of the ice picked up
But a flight from Goodwood to Prague, IFR, with a duff AP? I would not go. I would get the AP fixed. I would not accept the reduced safety of the increased workload. I would go with a co-pilot (and indeed I have done more or less exactly such a flight with a co-pilot a year ago, when a replacement for a failed pitch servo had not yet turned up from the USA).
Another reason I would not generally fly for real with an INOP AP is that unless the fault is positively diagnosed, one cannot be sure there isn't a latent defect in the electrical system which is actually not limited to the autopilot system. This is a much wider safety policy issue and is why I don't like flying the old GA training scene wreckage which is often covered in INOP stickers. Is such and such item INOP because the specific piece of kit is duff, or because a wiring harness is slowly getting cut through / shorted out against a bulkhead? A lot of defects are left undiagnosed due to working under the cost pressure of a typical FTO owner. Unless somebody chased down the defect, you just don't know, and "not knowing" is a no-go for me.