Look at the EGT!
Why on earth would a pack controller (a solid state electronic device) fail if used alone instead of as a pair? As Black sheep says it is possible you will reduce the MTBF of some of the pack componants (ACM for example) as they have to work harder.
However, your captain is incorrect. The most efficient way to operate the APU/packs (on the 75/76) is either with no packs AND the bleed turned off, or with BOTH pack operating.
The APU EGT is a fair measure of how efficiently it is operating. If the APU is running and the bleed is off the the APU EGT will be low (for a good APU below 300 degs C on a standard day). Switch the bleed on and the EGT will rise quite sharply (a 100 deg temp rise is fairly normal) then as each pack is switched on the EGT drops again. With both packs operating the EGT will be lower than with a single pack operating.
I believe Boeing recommended SOP is as per the above and it is the SOP in my company.
Hope this helps.