Well, joking aside, early on in the spring and late in the autumn, early november it might actually on occasion be a good idea. On not so serious note...it would probably only puncture and deflate my self opinion if I would take your advice ad notam and then fall over....
If you've ruled out squared tyres then I think it may be a little too unstable in the suspension settings. Maybe someone has adjusted things (ride height) to be too high compared to standard and has made it "nervous" in the corners. Have you changed the settings yourself? It might be a good idea to record the settings as they are now and then compare them to standard. If they are a long way off (say 25%+) then you could return them to standard and see how it handles after. As long as you note what they are set to now you can always change it back.
all of the above,plus worth checking rear wheel alignment(old fashioned way,two lengths o stick running length of bike to check rear is parallel with front wheel etc). too-soft front suspension/tyre pressures,and uneven fork damping can also cause the feeling of lack of control and instability at low speed(hi speed as well come to that). having your steering head bearing too tight will cause slow speed manouvering problems. but most commonly it is the tyre becoming v shaped.
I had a 907ie for about 10 years. Whilst the handling was never in the 851 and 916 league that I also had at the time, it still handled pretty well. Tyres would be well scrubbed in all over and I really enjoyed cranking it over in bends. But in the final year or two, the handling inexplicably "went off". It really felt as if the front would wash out in corners. I was using the same tyres (Dunlops at the time) as I had for ages, and all the suspension was looked at and nothing could be found. I finally sold it, because I knew I was going to crash it and no answer to the unconfident handling could be found. I suspected that the odd box-section frame had gone off (which shouldn't be your problem) but it remained a mystery. So could it just be an age/metal fatigue thing?
My bike was very stable and sow steering until I had the suspension set up, it took me a while to get used to it tipping in quicker but my point is that adjusting the suspension can make a big difference, might be worth checking the settings, it may be set up too soft.
Tyres good? Front forks right height? Rear shock/ride right standard? Air in the tyres? If all thats right its F*cked. I'll give you a tenner for it
Time for an update and an extra, thank you, to those of you that provided tips and insights. We now believe we've found at least one thing that might if not fully and totally cause this "falling over" behaviour. It turns out there's play in the swingarm bearings. What's odd though, is that it's more or less only detectable at a certain angle and only if you're strong enough! (It's back to the gym!) The wife participated in an on track rider's training day(not a regular track day). As part of the pre-track safety inspection, she mentioned that the bike was, well, a bit "agile" in turns. They -the group the wife was in- then collectively managed to manhandle the bike and the swingarm and detected/provoked the play in the bearings. There might still be other issues remaining but all the easy ones as tyre sizes we've checked and they're all OK. Again, thanks for your input!
