I think you are confusing a few things.
First we are talking mainly about the tagger link on MB.org here. If you click on this the release this tagger link was for should load in the right panel of Picard. If it is already there nothing happens. So for testing please make sure Picard’s right panel is empty. Then click on a tagger link and see whether the release in question gets loaded into Picard. If yes, good, if no please post the output of “Help > Show Debug/Error log” here.
Lookup in browser does not do anything inside Picard, instead it opens a browser window and tries to find the song on musicbrainz.org. Then you can search on MusicBrainz.org and load releases / recordings into Picard as described above.
You only need to use this if the lookup and scan functions inside Picard do give no or wrong results. Otherwise the usual workflow is to first cluster your files in the left pane, than try “Lookup” on those clusters, and for anything still missing give “Scan” a try.