The link to NYT URL above is I think a subscription page as there is a login at page top. Do you have a subscription to the NYT web account ? More information please.

I do have a subscription -- being logged in does not change behavior for me I checked with Safari -- you do not need to be logged in to play (at least some of ) the videos, e.g, https://nyti.ms/2G43nk5

REFRESH INSTRUCTIONS Please Refresh but do this 1st: Delete your Cookies and Cache and TEST. *https://support.mozilla.org/en-US/kb/export-firefox-bookmarks-to-backup-or-transfer *https://support.mozilla.org/en-US/kb/back-and-restore-information-firefox-profiles then *https://support.mozilla.org/en-US/kb/refresh-firefox-reset-add-ons-and-settings TEST....... If no issues then Extensions which need to be added back in 1 at a time and tested ..... Or it is your Profile : Make a new one and test ...: *https://support.mozilla.org/en-US/kb/profile-manager-create-and-remove-firefox-profiles If is your Profile : *https://support.mozilla.org/en-US/kb/recovering-important-data-from-an-old-profile *http://kb.mozillazine.org/Transferring_data_to_a_new_profile_-_Firefox Note: Any customization will revert back to default, you will also need to reinstall Extensions. Please let us know if this solved your issue or if need further assistance.