I am having the same issue. I have 4 controllers and 4 headsets, all of which work for a while, then sporadically and then not at all.
About a month ago, I spoke with someone at xbox support who said it was a software issue that the third party headsets were not updating and they had an update coming in July to fix this. They sent me another controller for free (thanks, but please just fix the problem). The new controller worked for about 3 weeks and then I am without a working headset again. I called back into xbox support and they didn’t know anything about the software fix they had promised me last month. After talking to a supervisor, they told me that xbox will no longer support 3rd party headsets. I let them know that this answer is INSANE, especially since you can still purchase 3rd party headsets (including the one I have) on the Microsoft store website. I asked for their response in writing or to speak to someone else, both of which were refused. I later spoke with someone else in tech support and they are proposing that it may be a problem with my actual xbox and I need to send it in for 3+ weeks to check it out.
Needless to say, I am not happy with xbox. I have a 7 month old xbox with 4 controllers and headsets and none of them are currently working.