I think that I know where is the problem of your calculation.

Your friend retirement budget is $20K a year.  To produce that $20K, he/she needs retirement asset of $500K with 4% withdrawal rate.  The 4% rule is a very common pratice in retirement financial planning.  The $500K asset is not being touched and it will grow since the investment return will be more than 4%. 

Your calculation is based on age.  First of all, your plan is based on a person's retirement budget and age to figure out how much he/she needs, which is impossible to calculate since no one will know how long a person will live.  Since no one knows the life expectancy of a person, a simple approach is applied by 2x.  Second, your plan is a saving and withdraw plan, not an income produce plan.  The following is some scenarios of 4% rule plan and your plan.

US life expectancy is 78.5.  The person is 60 years old.  Annual retirement budget is $20K

4% rule, he/she needs $500K, your plan is $370K.

Scenario 1: he/she was hit by a bus and dyed at the age of 61

       4% rule: $500K left to the heirs

       your plan $350K passed to the heirs

Scenario 2: he/she lives all the way to the life expectancy of 78.5

       4% rule: $500K left to the heirs

       your plan: nothing left $0

Scenario 3: he/she lives to the 100

       4% rule: $500K left to the heirs

       your plan: he/she ran out of money at 78.5.  I don't know how they live from 78.5 to 100.