I have seen other inboard auto-tuners that are not happy trying to match a low VSWR to a better VSWR. It shouldn't be a concern if the VSWR is already <1.5:1. Just leave the tuner turned off and let the rig do it's thing. You can't hurt anything, and the mismatch losses are minimal.
Ditto. I've seen this effect also. Seems the algorithms (code) in the controllers aren't always equipped to deal with an already-low SWR. Just turn the tuner off and use the rig.
I have an FT-950 and it will tune even with an already low SWR. I can see some programming telling the rig not to tune if the SWR is already low. But definitely not in my FT-950.
Now, I cannot recall for sure, but my Icom 746 may have behaved in the manner you mentioned.