In 2010, the Post Office stopped the requirement for including County in a postal address so it is indeed not necessary to keep it for the purposes of 'a complete address'.
Currently on the Post Office help system, the answer to the question Should I use a county name in my address? is:
When you address your mail, you don’t need to include a county name -
your letters and parcels will reach your intended recipient without
one. If, however, you’d prefer to include a county name, you are
welcome to do so.
Having said that, not everyone knows this, and many users may prefer to include it in order to increase their own confidence in the address they are providing - especially when they know there are multiple towns of the same name. Yes the postcode is the key differentiator, but still people like completeness and there's nothing wrong with providing a little extra information if users feel it's an important part of their details and if they feel some delivery of an important item is at stake.
For this reason it may be desirable to allow users to enter the County, but not to make it a required option, and for the underlying system not to enter the detail into the database.