The reason you cant just remove the avatar from the html is because it appears the content is built before the html file so its all inclusive you cant just remove one item per say from the html without removing the whole content. The avatar css comes from photo_floatbox.html
So what i did was i attacked this issue from the php side. So that the avatar data never gets passed to the links content array build in the first place. The only thing that is passed is a null like so ''..
Here is what i did and the avatar went away.
open ow_plugins/links/components/links_widget.php
scroll down about half way and you will see this line
$avatars = BOL_AvatarService::getInstance()->getDataForUserAvatars($authorIdList, true, false);
This line actually grabs the avatars data for the users
so all we have to do is just comment that out like so and then just make it null.
// mod keep avatars from showing on widget by loading data with null value
// $avatars = BOL_AvatarService::getInstance()->getDataForUserAvatars($authorIdList, true, false);
$avatars = '';
thats it... try it... Now the only thing i did not do was turn on debug to see if it was pulling an error or not (warning and notice is fine) if no actual error then your good to go..
Also remember that any custom mods you do need to be put back when an update is performed as it will overwrite the changes. So make a list of your changes always.