Out of the summer season, yes a good place to stay would be Corfu town although some other hotels, villas & apartments are open within resort areas and villages. As you quite rightly say, a lot is closed down, however there is still many places open. With a mix of winter residents whom live or other nationalities whom choose to remain for the winter on the island, it does offer options to explore Corfu for a non beach style holiday, villages tend to be busy during the winter months and many cafenions, tavernas and such are open and the atmosphere is fantastic, then there is Carnival Time and other festivities over the winter which will make ease of journey for those wishing to visit friends and family and of course vice verse for us to visit family and friends in UK.
The community (120,000 Corfu residents) and also the UK community has been campaigning for a direct flight for as long as I can remember certainly over 20 years. With many Corfiot teenagers studying in UK this opens a whole new life.
Choosing accommodation you should check there is a form of heating available as winter evening and nights can be cold & damp and of course a car is also useful, although many buses still run as this serves the local community and villages.
With the option of twice weekly flights a long weekend or a midweek break is now an option for a city centre holiday or just to visit friends and family here.