Just running a Honda (or anything else) until it stops does not fully drain the carburettor. If there is some E10 (or E5) left in the float bowl it will sit there attracting moisture, the ethanol attacking the aluminium parts, diaphragms, seals, hoses etc. and the rest turning into varnish to further mess things up. Hence my policy to flush through with Aspen. Unless the equipment is going to be stored for a long time I don't bother to then drain that from the bowl, or run the diaphragm carbs dry. I suspect, if anything, it keeps the "rubber" parts in good condition.
I do likewise with my outboard motors. Which, like my generator also have diaphragms in their fuel pumps (not integral with the carb., unlike the chainsaws etc)
Dealing with a clogged up carburettor/rotted diaphragm etc. is very tedious, time consuming and expensive. Sometimes a new carburettor is the only answer.
Before Aspen came out I used to use "fuel stabiliser" in the hopes that it worked. Maybe it did, I dunno.
Want to know if your older car, bike etc. is compatible with E10 ? Try looking here
Check if your vehicle can run on E10 petrol
Now, where does this (bio) ethanol mostly come from that now makes up 10% of our pump petrol ? I suspect very little from the UK.
Common ethanol fuel mixtures - Wikipedia