In all the plans that I have seen the Employer is the Plan Sponsor and also the Plan Administrator, whereas any TPAs (record keepers, Trustees etc) are service providers.

Can anyone explain why this is done? Why is the TPA not the Plan Administrator?

Any cites and links to explanations would be very helpful because I need to explain this to a client, not only as being standard practice but why.

The main reason is that the TPA does not want the Fiduciary repsonsibility of being the ERISA Plan administrator.

You may wish to search prior threads about naming the employer as plan administrator. In one of them, Kirk Maldonaldo explains that technically, a plan administrator is not a fiduciary if the plan administrator is limited to only those record keeping and reporting functions expressly assigned by ERISA to the plan administrator. His is a nice analysis and probably correct, but it requires a precision and discipline that most plans cannot maintain. A plan administrator is likely to slip into fiduciary functions and it is also practical to have the plan administrator be a fiduciary. The fiduciary would have to oversee a "pure" plan administrator."

The employer should never be named as the fiduciary and you are correct the the TPA will not want to be a fiduciary, so the TPA is very carefull to avoid "plan administrator" or any other label that implies "fiduciary."