Can a UK business appoint its own Responsible Person?

Technically yes, practically rarely worthwhile. A Responsible Person must have a place of business in the EU or be able to operate there. If you're a UK seller without an EU office, you'd need to send one of your staff there or hire someone new — that quickly costs more than hiring a service provider. Moreover: the regulatory requirements demand specialised knowledge that external service providers have routinely. Most successful UK sellers therefore opt for a dedicated service rather than an in-house solution.