I have holed out from the fairway many times for eagle, and also eagled a par 4 for a 2 one time. The eagle on the par 4 occurred during a solar eclipse so I will never forget that one. With all this being said, I would 'definitely' rather eagle every year once or twice over a hole in one, btw, in which I never have had one. A friend of mine who got a hoi a few years ago said it wasn't the dramatic experience he thought it would be getting a hoi, just 'lets play on' sort of feeling.
We only have two parallel holes at our home club, close enough so you can accidentally hit into the adjacent fairway, 1 and 10. However, we have a couple more blind tee shots, occasionally we realize someone is "in play" while a ball is in the air.
Like @iacas, I'm hesitant to yell when it seems clear that the ball will miss people. I've seen enough folks move in unexpected directions when they hear "fore", I don't want someone to go the wrong way when startled. I've also seen too many start looking for the ball coming at them, which scares me. I turn my back to the shout and cover my head. I'd much rather take one in the back than in the face.
I'll admit, I've never timed a search, so I don't have data about the impact of the rule change. But I do know that we're made an effort to terminate a search earlier than we once would have, so that's positive. I don't have a real problem knowing that during the COVID times, the Tour professionals will have difficulties similar to mine when they hit errant shots, I have very little sympathy. If you want to find it, hit it where there's short grass.
There's no "right" answer, but I'm fine with 3 minutes. I don't think that too much shorter is appropriate, and I wouldn't recommend going back to 5 minutes.