Lacey is entitled to his opinion but IMO he is over-egging the "diss" of Archie not being pictured on the desk.
He minimizes that as The Queen verbally mentions birth of Archie, the photo taken a few days after Archie's birth featuring The Queen, Prince Philip, Meghan's mother Doria, Harry, Meghan and Archie flashed on the screen.
He states:
"It was unheard of for the royal Christmas desk not to feature a cosy image of the latest royal [emphasis mine] grandchild or great-grandchild. "
This is an incorrect statement as Archie is not a royal great-grandchild. Now if there were pictures of the all of Peter and Zara's children on the Christmas desk the years of their births then he would have a point. The thing is that he states that it is "unheard of" but there is not really a precedent to match Archie's circumstances, which is a great-grandchild who was born non-royal and may become royal at a later time.
Lacey goes on to itemize some actions by the Sussexes that were relevant but he couches it as the Sussexes were "non-personed" due to The Queen's "displeasure" with them. And while it would be naive to think that she was not displeased with the Sussexes in December 2019, IMO it is not an unreasonable course of action to not include a photo of a non-royal grandchild whose birth was shrouded in misinformation, whose parents refused to give out the names of his godparents and whose parents reportedly want to raise as a private citizen. But it should be again noted that Archie's arrival was mentioned during the broadcast accompanied by a picture.