I cannot believe that fans are upset with Discovery using the Delta. After all, in canon the Delta is not the Enterprise insignia but is in actuality the 'The United Earth Space Probe Agency' (UESPA) exploration insignia which was later incorporated into the Starfleet emblem.
Here it is on the Friendship One probe launched by the UESPA in 2067:
(The Delta had been amalgamated with the United Nations emblem, but is clearly the Delta)
And again in 2155 where the UESPA Delta had been incorporated as part of the Starfleet Command emblem:
In canon we have seen the crews of two pre-TOS, prime universe, starships use the Delta.
The USS Franklin and the USS Kelvin:
^ Like it or not the above is canon
In the TOS era we see that some of the Constitution class ships and a few starbases had begun to use their own unique insignias. However as far as starships go perhaps only the Constitution class crews were allowed to diversify their insignia, the rest of the fleet may have used the Delta. The Enterprise is the flag ship therefore they may have chosen to use the Delta in solidarity with the rest of the fleet. It should also be noted that the UESPA was the primary operating authority of the five year mission which the Enterprise undertook, as Kirk explained to John Christopher in the episode "Tomorrow is Yesterday". Therefore we should not be surprised that the Enterprise uses the UESPA exploration Delta during its five year mission.
Having said all that, Discovery is set at least ten years before TOS so perhaps all ships at that time, including the Constitution class ships, all used the Delta. It is only by the time we reach TOS era, a decade later, that some ships have begun to diversify their insignia's. However, this was only temporary as in TAS all ships had begun to use the Delta once more.
Granted, the new command logo does include the MACO five-pointed star (that is what Memory Alpha calls it!), just like the Enterprise crew wears. However the Command/MACO star is not unique to the Enterprise either.
I know people think that using the Delta in Discovery is lazy writing or something, but based on canon I cannot see any logical reason why they should not use it.
