For my Nepal trip and climb of Imja Tse (20,305), I trained with weights and ran 3 times a week. It really helped to be in good shape and not struggling with cardio. I did a lot of long day hikes in the White's also.
My travel nurse tested my reaction to diamox prior to going on my trip, as I have allergy issues. No one in our group took diamox, but a few had to go down to lower elevation for a few days. Some of them didn't attempt the climb. ( it was assuring to know I wouldn't have a reaction in case I needed it)
I had a severe headache on a 17.5K climb during the acclimating and had to rest that evening and pump the Advils' and lots of fluids. I was fine going up but got the headache on the climb down..3 of us got the same explosive headache.
In the morning the headache was gone and after a day I was back to normal and suffered no more issues. I think drinking tons of fluids and eating often and yes going very slow..is the best thing you can do for a sucessfull summit.
In the long scheme of things...it's not just the summit but the whole experience that counts!
The better condition you are in, the better the experience will be..