Step 1: Separate "commercial banking" from "central banking":
Commercial banking means taking deposits from and lending to the general public for profit. Central banking is an entirely different, non-profit, regulatory/apex role in the financial system.
Step 2: Show the central bank IS still a "bank", just for different clients:
It holds accounts and extends credit — but only to the government and to commercial banks, not to individual citizens, which is why it is called "banker to government" and "bankers' bank", not a commercial bank.
Step 3: Conclude on the original claim:
So the fact that the public cannot open an account there correctly shows the central bank avoids retail commercial banking, but it is inaccurate to say it does "no" banking activity at all — it just directs that activity toward government and banks instead of the public.
Final Answer:
Correct in the retail sense, but the central bank still performs banking functions for government and commercial banks, plus unique functions like currency issue and monetary control.