Oh and here's the output you should receive:

Installation Result

   Message: The update completed successfully, but the system needs to be rebooted for the changes to be effective.

   Reboot Required: true

   VIBs Installed: Intel_bootbank_net-igb_5.2.10-9

   VIBs Removed: Intel_bootbank_net-igb_5.3.1-1OEM.550.0.0.1331820

   VIBs Skipped:

That is installing the modified igb driver into the latest 6.0u1 build 3380124 (This month).  I've also installed this modified build back to v5 with zero issues for this exact issue as well.

Your issue is clear right?  Hardware seen, driver not loading.  That's what the lspci is showing you above.  When the driver attempts to load, due to the NVM checksum the driver looks doesn't like the checksum and just refuses to load.  The only thing modified in the driver link I posted is to simply bypass that NVM checksum check - no other driver changes so should be zero impact to performance, params, etc.

Don't feel bad this had me pulling my hair out for quite some time too before I found the solution.  The good news is, rolling back to older esx builds is NOT needed here - you can absolutely run the latest and greatest with this NIC!