How to make the Jenkins agent build then start the server?
You are running into a conflict between jenkins nature of making sure everything is done before a job is complete and your desire for some long-running things to be asynchronous.  There is no right answer here, both are reasonable desires, but you and jenkins are pulling in different directions.  It may be ok for some parts of the jenkins job to not be completely done but they are good enough for the job.  This is a decision you get to make.

If getting this working inside of jenkins is too much trouble there are work arounds.  You can make your own daemon that looks for job completions (via files or database entries) and then starts something else.  Obviously this means you lose the UX features of jenkins that let everybody know where something is.  But you could make this work inside of jenkins by having different jobs also.  The quick job finishes and then the longer job takes over.  Since each is its own job it can have timeout requirements that make sense for it.