There is no hardware limitation in the Switch for not supporting bluetooth headphones, because the original Switch already has a bluetooth 4.1 compliant chip. Technically, this is a driver (software) issue. Logistically, this is a management and corporate bureaucracy issue, because all that needs to happen to remedy this is that a corporate higher-up needs to command one of their countless hardware subsidiaries ( https://en.wikipedia.org/wiki/List_of_Nintendo_development_teams#Subsidiaries
) to finally implement bluetooth headphone support for the Switch bluetooth driver. Nintendo could literally add support for bluetooth headphones via a software update, by simply hiring a full-time programmer to write the driver, a programmer to implement relevant driver UI stuff to the OS, and a tester to test out various common bluetooth devices, and they could finish this in a few months, or quicker if they hire an extra programmer to help out with the driver development. The PSP Go from 2009 has bluetooth 2.0 support and fully supports bluetooth headphones.
The good news for Switch owners is it looks like they may already be taking steps in finally implementing headphone support based on this recent bluetooth driver update:https://www.pcmag.com/news/report-latest-switch-firmware-includes-bluetooth-audio-support