I would say no - really bad idea . Firstly the dog is not a pet so there has to be enough separation to make wise , absolutely necessary decisions about the dogs suitability and progress. Many dogs are washed out along the way .
The pet issue , a pet is a pet . A service dog comes from a sophisticated breeding program , a sophisticated critical evaluation program and will be tested for performance and ability, including temperament and ORTHOPEDIC health , all along the way. Then at the end THAT dog is matched with the perfect recipient. Even here it is not just the first person , sometimes a few changes in handler need to be tried - reasons could be anything from the handlers physical ability, the dogs stride , the dogs energy and power . An 80 year old woman is not going to get the same dog that a vision impaired 25 year old man would get (more or less)
Secondly the institute has accredited trainers who know what they are doing, a chain of personnel to be responsible to , and INSURANCE for liability.
Without the "school" being involved you would never get public access to enter those places where only a service dog or dog-in-training may enter , legally.
"sad to me that a foster family raises the dog and then hand her/him over for training" the foster family knew very well what they were in for . They get a sense of pride and accomplishment to send their protege out in to the world -- and then they get ready to do it all over again, and again.
"All dogs get attached to their families, and GSDs seems particularly sensitive to separation."
A dog with separation anxiety is not a good working candidate. We (I) do this all the time with every service dog that is prepared for service. Dogs that have had a rich rewarding human-bond attachment will seak out the same experience when the next guy comes along. In the raising of guide service dogs the dog is not treated as a pet , right from the beginning .