I have the same. I wanted to start BF V as I have daily for the last few weeks without a hitch, and now EA Desktop says I need to install it, but doesn't give an install/download button, only an option to buy another edition. I tried all fixes with 2 EA reps, to no avail. A few games do show the download button, but at least BF V, SW: Battlefront 2 and Titanfall 2 do not show the button. SW Squadrons DOES show the download button.
I have EA Play through XBox Ultimate Gamepass, so I can't switch back to Origin to play, as in their infinite wisdom EA decided to have this quite big feature limited to the EA desktop beta-product.
BTW I had all of the games I mentioned above installed, but when EA desktop couldn't find them anymore (no disk changes of any kind), I reinstalled EA desktop - which without any warning or even a casual mention deleted hundreds of GB of installed EA games. If you HAVE to have your own game launcher (and trust me, not a single player loves having to have multiple game launchers installed which all do essentially the same), please at the very least keep basic game launcher functionality intact, like leaving the game installation folder alone when uninstalling.