I have flown it many times, and would also not go west of Canary Wharf in a single.
It all comes down to your interpretation of 'alight clear'.
From 2000-2400 ft there are plenty of places to which you can glide to a reasonable chance of a successful landing in the average light single (especially with a westerly wind). One of these is indeed LCY. Whether or not singles are accepted usually, I would argue that in an emergency you can go wherever is best, and a concrete runway seems a better option than some waste ground. I seem to remember a Rockwell Commander ran out of fuel a few years ago and put it down near LCY. Didn't he get 'done' for his lack of fuel planning but not for flying over there in a single? (The same aircraft crashed last year killing all on board).
If you mean: 'Alight clear of the whole conurbation' then you are talking 6000ft plus, but then the same is true of many places we fly. The 1500ft rule over a congested area does not of course relieve us from the need to 'alight clear' but how many people can honestly say they have ALWAYS been able to alight completely clear of Milton Keynes, Bournemouth or any other reasonably large conurbation they have encountered on route?
If it's OK for single engined helicopters to be able to use the Thames in the case of emergency, why can't fixed wing also choose to dump in there 'in extremis'?
To summarise, therefore, it is my opinion that provided you are sensible and fly as high as the clearance or cloud allows, then you have a good chance of putting it down without endangering others than yourself.