"All three banks were super-easy to deal with, they made out checks payable to Vanguard FBO Client, and he mailed them to Vanguard,"

From the IRS:

"IRA trustees can accomplish a trustee-to-trustee transfer by transferring amounts directly from one IRA to another or by providing the IRA owner with a check made payable to the receiving IRA trustee."

Yay! It's important to never have them payable to the taxpayer, in this scenario.