Universally verifiable random beacon
There is no sensible solution to this.  It is impossible, even if this was not a hypothetical question.  It cannot be done for primarily two reasons:-


You cannot have the nodes measure any analogue quantity.  Analogue measurement noise will govern the accuracy of the reading.  Coupled with the typical hash based randomness extractors, the avalanche effect will guarantee that every node will generate a different random number.
So you're left with digital entropy acquisition.  Lottery results are good but there's not a lot of them to get the few bits /hr (maybe just - you'd have to do the math).  This was done for the Million Dollar elliptical curve. Much larger quantities of digital randomness can be obtained from international market information acquired via the appropriate APIs.  This will gradually fail though.  Spotty developers will change the access APIs every 18 months.  The nodes will gradually lose access to the digital entropy.    This gradual degradation of sources will affect the nodes no matter what sources they use.  The API degradation issue would also affect the lottery technique. And.  You can't update the nodes as that would put them under singular control breaking your initial premise.


The only kinda solution (breaking many requirements) is to use a distribution network sending out randomness like the NIST Randomness Beacon.  But it would have to be under international /UN control and inspection and outside of US jurisdiction.  It can't be NIST itself as that is a politically controlled arm of government (US Department of Commerce.)  Or do the same for updating the nodes with new market access APIs.  If the nodes were open hardware /open source based and not sealed, it would then be possible for hackers /scientists and journalists to verify their on going correct function.  But that would then allow prediction of the random numbers as the entropy and algorithm would be known.