Welcome to the Community,
please take a look at https://www.v-front.de/2015/03/vsphere-6-is-ga-ultimate-guide-to.html which provides information about adding the required drivers to an ESXi v6.x image.
if you take another look at the output, you will see, than another card with the same class is working – "D-Link System Inc DGE-560T PCI Express (x1) Gigabit Ethernet Adapter [vmnic2]" with Class 0200: 1186:4300.
And "D-Link System Inc DGE-528T PCI Gigabit Ethernet Adapter [vmnic0]" with Class 0200: 1186:4300 doesn't work, ESXi doesn't see it.
There is no any other driver than Net55-r8168 for 1186:4300 (or specifically for DGE-528T) and Net55-r8168 was already added to ISO.
the driver is loaded for the DGE-528T card, and it should show up as vmnic0.
Since it is not listed by "esxcfg-nics -l" there was probably an error when initializing the card.
Can you please check the host's boot log for related error messages?
In an ESXi shell run something like
gunzip -cd /var/log/boot.gz | less
to look at the file. Search with "/vmnic0".
I checked this log on my side, and just one entry there saying:
..../vmnic0 named as "" .