stem height

alex_van_niekerkpwo5l_Wc

for years now I'm riding a mountain bike as a road bike. I used to have a road bike but it did a number on my lower back (and yes, I do have sufficient core strength). I want to go back to a road bike because, face it, it's much faster. My Question is, will the 3cm extra stem height make the difference for me to go back, or do I need a higher stem? And which would be the best road bike for that?
Thank you