What is the money talks opinion on this?
If the portfolio manager is very good and the underlying funds are good funds with good portfolio managers than it is a great strategy. However, if the funds are sub-par or the portfolio manager is sub-par than it becomes a bad strategy.
Does one have to pay fees for TRSGX and also for the funds that TRSGX invests in? Basically, is the company getting twice the fees/income by taking my money, which I've invested in TRowePrice, and investing it back into a TRowePrice fund?
You are hit twice on fees but not in a sense that you see. The fees for underlying funds are just reflected in the NAVs, same as the fund of funds.
My advice is to just make sure that the PM is good and the funds are good too. If they both are, you end up getting "layered alpha", where you have underlying funds that beat their benchmarks and the fund of funds that beats its benchmark. But this can also work in the opposite direction as well.