The short of it: It's Vsync causing the issue, along with external framerate limiters. As long as vsync is disabled, the weapon/power pause after using powers/melee doesn't occur. If you're fine with vsync disabled but still want to limit framerate, you can use the game's own 60/120/240 limiter, and it does not cause the problem.
But, enable vsync in the game, through your control panel, or even by using something like DXVK, it comes back. Keep vsync off and use something like Rivatuner to lock the framerate, it comes back. Have to keep vsync off and only limit the framerate from within the game options to never see this bug.
I'm making a video, when it's uploaded I'll make a new thread with a link. Doubtful EA will patch it by now but at least it will offer some explanation if you've run across this and are bewildered why your character will randomly stop for 3 seconds after chucking a geth across the room.