Where do HMRC say that? My understanding is that it is the employer's responsibility to operate PAYE correctly and that the amount paid to the employee is always treated as being net of deductions, which are the employer's responsibility to pay to HMRC.
It's actually a little bit of both! Yes, employers have a responsibility to operate PAYE correctly but employees also have a responsibility to ensure that the look after their own tax affairs - this includes through employment. As an example, a company a worked for a few years ago had an employee who also had pension income. Due to an error in the use of his tax code he was "under taxed". When HMRC discovered this they wanted the tax paid by him, through his tax code.