Is the DOJ case against Google stupider than the FTC's tech cases?

The Dept of Justice (DOJ) is upset that Google pays Apple to be the default search engine on iPhones.  If the DOJ wins their case, the "relief" DOJ would seek would proscribe such payments, just as the EU did.  According to one observer, 

"In the wake of the European Commission’s 2018 Android decision, Google had to implement a choice screen (starting in 2019) on Android devices in Europe (which had confined the market to Android devices). No more Google default: users of new devices with the choice screens are presented a half-dozen choices—including Google and obvious alternatives—upon startup. Placement is shuffled at random."

Did this relief change Google's "illegally" acquired & sustained dominance in mobile search?

Well, not exactly.  Google’s share of general search on mobile phones in Europe is still greater than 96%.

QUESTION: So, why would Google pay for something that it would get anyway?  

TENTATIVE ANSWER: Apple has valuable real estate and they are creating ex-ante competition among search engines for the right to use that real estate.  But if search users are going to switch to Google anyway, even if, e.g., Bing outbids Google, then maybe the ex-ante competition is not very fierce.  I wonder how much Google makes on search ads served up on Apple devices, and how much Google pays Apple for the privilege.  

QUESTION: And why would the DOJ ask for relief they know will not work?

TENTATIVE ANSWER: DOJ doesn't care about ex-ante competition.  The see Google's big share of search after they won the ex-ante competition, not as a indicator that the  the most valuable search engine won the auction, but rather as an indicator that something is wrong--or at least could be made better by government interventiion.