In section 25.1, sub-section "Credit Default Swaps and Bond Yields", of "OPTIONS, FUTURES, AND OTHER DERIVATIVES", John Hull defines "CDS–bond basis = CDS spread - Bond yield spread", and claims "The effect of the CDS is to convert the corporate bond to a risk-free bond (at least approximately). [...] [This] suggests that the CDS–bond basis should be close to zero.".
I cannot find a rigorous justification for this. Let's take a simple example: maturity = 1 year, bond APR = 5%, paid twice a year, CDS rate = 2%, riskless rate = 0% for all maturities. The claim would then mean "Bond + CDS rate = 3% Treasury", and given the CDS is free to enter (issued at par), then that would mean the bond price is 103.
Yet, if we try writing all the cashflows with simple assumptions (recovery rate = 40%, only possible default times are t = 0 and t = 0.5, and payments are made in arrears), then we have the below cashflows:
We see CDS + Bond still looks rather different from a one-year, 3% treasury, because the whole position can be extinguished at t = 0.5 in case of an early default, and you would then need to reinvest the proceeds, so you could never exactly replicate a "bond + CDS" position with treasuries only. In fact, in this example, CDS + Bond always has cashflows lesser than a 3% treasury, so the value of the credit should definitely be less than that of a 3% treasury, so the "CDS bond basis" would definitely be positive.
What is wrong with this reasoning? Given these caveats, why are we paying so much attention to the "CDS bond basis"?