I do offer help, tending to suggest jobs rather than 'is there anything I can do', but I wouldn't dream of muscling in and doing anything not asked for! For instance, my 92 year old neighbour has been very poorly and weak with shingles, so I have offered bed changing, laundry, cooking, driving to GP, shopping etc. She is usually very independent but is now comfortable with requesting these things.
I am also fiercely independent and can be very difficult to help! The only help I would seek if ill or hospitalized would be cat care. I have a printed instruction sheet with the supply of their food and medication just in case!