Hmmmm...interesting discussion. I have been contemplating putting a LH safety on my Win M70 to try and all that before reading this. I find that on rifles with traditional, swept back grip I simply automatically wrap my thumb around the grip. I think the thumb "off" the grip (only resting on top side of it) is more of a match shooting thing and especially with vertical grips (as on target/tactical stocks) in order to prevent torqueing the grip and inadvertently canting the rifle and/or affect the follow-through.
Either way, familiarity and muscle memory with whatever safety you use is key. I suffered from having new to me CZ 550 with safety opposite to ZKK rifles I was used to for couple decades. I thought after years of not owning a ZKK I would simply adapt to the new. Besides I always have different rifles with different safeties. Yet in a heat of the hunt (and this was simple plains game hunt), I pushed safety forward as I got on the sticks and got ready to shoot. The animal went behind a bush and stayed for a while. After few minutes that seemed like eternity, with me still on the sticks, the PH and I started discussing whether to move or wait when the animal appeared, moving from the current position to another bush, giving us a small window. PH whispers: "Shoot.". Instinctively, just before setting the trigger, I pulled on the safety to "make sure it's off". I still remember my surprise when it moved. "But whatever", I thought, "on to the shot"...set trigger, squeeze....click! Surprise. PH again: "Shoot." Set trigger again, click! PH is looking back at me wondering what's taking so long, animal almost at the next bush. I rack the bolt, thinking chamber was empty. Live round is ejected, new one fed in. This time PH is like "WTF? Shoot!!!". I decide to just squeeze the trigger to speed things up, nothing. Note even click. That's when I realize safety is on middle position. Push safety forward, PH is now really upset "What the hell are you doing?!", I aim again, squeeze, bang! Animal goes down but the shot was not perfect, animal falls behind the second bush. We question the shot. It takes minutes to get up all dazed but alive still, yet we have no shot until he is much farther up the hill. Take another shot, see bullet deflect off a branch and dust off many feet to the side of the animal. Now get really nervous. Experience "hunter shakes" for first time in my life. Not liking this hunt at all. Animal moving beyond 400m, borrow friend's 7mm mag and nail it in the hip to stop it from cresting the hill. Animal crests the hill. We hike up and find it on the ground, 30 m over the crest, thanks to that last shot. Finish it off with a heart shot.
That's when I realized I needed way more trigger time with that rifle to burn the new safety lever motions to my memory. I also decided I would not use set triggers while hunting (I did once since but am moving away from them as cool as they are) and started thinking about simple 2 position safeties that don't let you open the bolt when on safe, etc...
Regardless, the safety was not really the problem, but familiarity with it under pressure was. And in my case I started relaxed but pressure mounted eventually and gradually and the set trigger added to the complexity and confusion.
My problem is having too many different rifles with different style safeties and triggers, hence one of my other posts on trying to dwindle down the collection and/or trying to figure out what I like most and/or convert the safeties to similar. When hunting at home or at matches, I do not use safety much, round under the bolt or not in the gun, and when in position and ready to shoot, I rack one in. I gotta find one way of doing things. The problem is I'm more of a gun nut that also hunts rather than a hunter who happens to be a gun nut.