Good day, I have the same problem here:
When I upgraded Moodle to version 8.1 I have lost the "link to a learmning object" as well in any browsers.
Which version of Moodle are you using?
When interacting with the DOOR database itself to upload LO,I cannot see that "Browse Button" as well using web browsers : Safari and Firefox.(MacOSX 10.4.9 it is working with windows) I have to use Opera and with this browser the "Browse" button for choosing the object is showing up
A discussion about this is also going on at sourceforge: