I just wanted to chime in to add some additional info about this issue. I did not know this was a problem until I found this thread when looking into issues with EA Desktop and mods for frostbite games which is not related to this issue. (I did find a tool that lets mods work anyway for MEA and DAI with EA Desktop)
However out of curiosity I did check my installation of Dragon Age Inquisition through EA Desktop and did find the Trespasser DLC listed but saying I do not own it just like other people here have reported here. However, in Origin it comes up as being installed properly and the game does launch fine through Origin. I though because I installed EA Desktop to take advantage of the EA Play and Microsoft Gamepass for PC combo, that EA Desktop somehow took over the launching of all the Origin games, however I found that is not the case. Only if both launchers are not running do these games default to opening EA Desktop to launch if both Origin and EA Desktop are installed. If EA Desktop is closed but Origin is opened the games launch fine through Origin when launcher through it.
Long story short, maybe a temporary workaround is to launch the game through Origin until this issue is resolved. As long as the game shows up and the DLC shows up properly like it did for me. Although my situation might be different since I have owned the game since launch and then later added the GOTY edition when that became available about a year later. I do not know if using Origin as the launcher would work for people trying to purchase it new now.
If not, I least I have contributed with confirming that this is affecting even people with the games being in the library for years.
I guess also this would also not work for people using EA Play through Gamepass for PC for DAI, as this really only seems to work through EA Desktop. Which is the only reason I installed it. In Origin the EA Play games do not show as being able to be played without purchase.