I had an intermittent overheating issue as well, fans came on most of the time but occasionally they wouldn't kick in at all. First time it happened the car overheated whilst being stuck in traffic, moving very slowly - unfortunately I was driving to a cottage I had rented for a holiday!
Up until this point I had no overheating issues which I had noticed as generally I don't leave my car running while stationary (don't get stuck in traffic often after a long drive!). Called the RAC, went through the same diagnostic procedure as you and couldn't find an issue - checked the blockages, fans kicked in etc.. I continued on my journey after the RAC guy left and the temp shot up several times, albeit intermittently, if the car were stationary for a small period of time.
Once I got home, tried the self test and one fan came on but the other didn't, tried again several times then they switched which one worked or just didn't come on at all. Fuses seemed to be fine as they both had turned on at some point during these tests so I figured it was likely the fans. The fact you said you had tapped them and they came on, makes me think it's similar.
Decided to take the OEM surround out and remove the fan motors and open them up. Large carbon deposits inside were causing the carbon brushes to be held back in some cases and in others the brush springs had to be stretched slightly as they were not touching the communicator at the centre. Gave the communicator a clean and removed most of the carbon build up inside.
Once I'd sorted both fans - worked like a dream without any overheating at all so far since. Granted most people would probably just replace them and if they fail again for me then I will
Long story short, run the self test several times and wait for the fans to kick in - ensure that they do every time. Also make sure to chuck some water on the windscreen or lift the wipers prior to that check
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