Or you do the test yourself, and just show the already-completed test to the guy at the office door, and keep the test hardware, just like you currently show but keep an ID badge.

Sure, some people could fake the test, but for this approach to work, all that matters is that most people don't fake it.

Perhaps, but if you think about how systems like this work today, for example scanning ID at the entrance to a nightclub or bar.

One way something like this is likely to be implemented is by validating a ticket or access card/token with a saliva swab. This is just too easy and attractive an opportunity for data collection to be passed up by some operators with business models that monetise the data as a revenue source.