Here is a Fed paper explaining QE. The authors note that the purchases of Treasuries were conducted by the FRDB of New York in the same manner as it conducts regular Open Market Operations (pp. 9-10). The New York Fed bank holds auctions for primary dealers, who sell the Treasuries to the Fed.
The article notes that the New York Fed Bank posted a summary of its QE purchases on its website as the program went along (p. 10, note 17).
To determine whether or not the Fed was shrinking the time between the sale of the Treasuries by the Treasury and the purchase of the Treasuries by the Fed, one would have to compare the normal OMO purchases before 2008 with the QE purchases. Given the Fed article’s description of the program, there doesn’t seem to be any evidence that the time lag shrank or, at least, that the Fed purchases were immediately after the Treasury sales as your friend claims. (One could go through each purchase on the website to see if the Fed made any immediate purchases.) In any case, your friend is correct that the Fed purchases of Treasuries, whatever the lag happens to be, provide lower-interest-rate-than-otherwise funding for the Treasury, which spends the funds on general programs of the Federal government. But the Fed indirectly funding the Treasury expenditures in this way is always the case for all Fed purchases of Treasuries and not a special consequence of the QEs.