I think I had similar experience with some of our grapes that started root growth and leaf growth and then fizzled - that happened to a few of those 4 but I am holding on in case they can make it anyway. I think I neglected watering the nursery one or two days too late after it got majorly hot and these smaller than 1 gal yogurt containers dried out faster than the 1 gallon pots.
What kind of varieties are you having the best success with? Are some kinds easier to start from cuttings?
I couldn't tell you the answer to that.. I am very new to grape propagation and know next to nothing about the cultivars.
Maybe try more cuttings next year for better chance of success and keep it somewhere shady/cool until they get going?
Yes, that makes a lot of sense. I know with apple scions, they recommend refrigerating the cuttings until it is the right time of year.
When I do my grape pruning is usually in January, when it is fully dormant, and freezing temperatures at night. By leaving the scions dormant in a cold place, maybe it will help it to build up more energy for roots in the Spring. Last year I tried planting some outside in pots with sand and rocks, but they just seemed too weak to adapt to the change of season. And the ones I forced to start indoors did not much like the lack of sunlight. I kind of think Late April, or Early May might be the sweet spot for planting cuttings.