Yeah but "if there is one Star Trek show ST09 and STID are proudly referencing", it's TOS.
They're not REFERENCING the original series, though. They're partly copying it but mostly replacing it. A few elements carry through, but it's not a reference so much as a replay with extensive modifications.
And ST09 and STID have, between the two of them, made more references to Enterprise than all of the other Trek series and movies combined. That's at least one reason to believe that ST09 branches off from THAT universe and that Archer's timeline was separate from TOS all along.