Funny, I am the other way around. I set my GPS to ping as I approach way-points and the off-course alarm to tell me if I have taken a wrong turn. That way I can just walk. I will sometimes put 3 or 4 way-points close together if there is an easily missed turn coming up so I get an audible warning to look out for it. That combined with the prior map reading while I am planning my walk means I often walk without referring to the map at all, let alone getting out the compass. Likewise my GPS screen time is very short indeed. The unit lives on my rucksack strap so it is a one handed operation to remove it glance at it and replace it. Absolutely no question of walking head down as you put it nor walking along with a hand full of map.
I always hated having to faff with the map, unfolding it, refolding it, breaking my rhythm while I stop to work out wear I am, getting out of the map holder and refolding it when I go over the page etc. An even bigger nightmare if it raining or blowing. I would say that having a GPS has markedly improved my walking experience.