Why is Homebrew lynx configuration not in my home directory
It's by design. According to the Homebrew homepage:


  Homebrew won’t install files outside its prefix and you can place a Homebrew installation wherever you like.


According to the lynx man page:


  LYNX_CFG            This variable, if set, will override the default
  location and name of  the  global  configuration  file  (normally,
                         lynx.cfg) that was defined by the LYNX_CFG_FILE constant in the userdefs.h file, during installation.


In some linux distribution, the constant LYNX_CFG_FILE in set to ~/.lynxrc (as I remember) but in Homebrew  this constant is left to its default value. You can check it with brew edit lynx. If you want to change the location of the lynx configuration file, you have, as you tell, to set:

export LYNX_CFG=~/.lynxrc