I grow a lot of grammatophyllum. I have at least 50 in inventory.
They will grow as fast as you give them more room. They take high sun more than Cattleya and lots of water and food.
I raise mostly Gramm scriptum and its hybrids that stay relatively small (3 foot high with 5 foot flowers) I find I need to re-pot them 3 times a year to allow room to grow. I have taken a 5 inch pot to 12 inches in one growing season.
I would never have the courage to a speciesgrow from a flask. You have a long wait and a lot of work. The New York Botannical garden took 15 years from a mature plant to get flowers from a Gramm speciosum, and I do not believe it has flowered since then.
For your original question, potting medium, mature plants grow in anything. Grammatophyllum have two types of roots. One type is like other orchids and grow down and the second type is much thinner and grow straight up. The vertical roots catch leaves and debris feeding the plant as they decay and building its own basket.
I grow in both bark and have others in sphag (the plant will eat the sphag over a couple of seasons, a little longer to consume the bark - and basket). It does not seem to make a difference.
They do not like cold and lose leaves when the temperature goes into the 40s but the plants survive without leaves into the low thirties, growing new leaves in the spring.
They seem to look best in baskets and you can let them grow out of the basket. I have several wher ethe air roots hide the basket.
gramm scriptum and its hybrids bloom regular and are only big not impossiblely large.