i went the opposite way and am still getting used to brifters circa 2015. before that I never had them and found myself pawing at my downtube once, twice, then going. $*&^ its brifters stupid! at myself. that said I do prefer indexed rear because you can just "throw" the lever into the right gear in an instant - whatever gear you want - so I shift quite a bit. friction is harder but I only have that on one bike at the moment and probably do shift far less just because of that.

as for riding with no hands, some bikes are squirrelier than others. but you should be able to just let go and the bike go straight - if you are wobbling its actually what you should do- your input on the bars actually can make it worse. I suspect that its not your hand off the bars that makes you wobbly, its the hand that is still on the bars. so try making that hand as loose as possible as you shift.