SSIS Script Task doesn't appear to run when scheduled
I've resolved this. I installed the SSDT Feb update and targeted the version to SQL Server 2014, as suggested by Dave. I did have to tweak one script (unrelated package and script task) but then I managed to build and deploy. It's processing happily.
The Pre/Post Execution messages were all clean, but based on the advice on this question, I did analyse them in more detail so I've picked up a bit of knowledge here too!
Thanks for all the feedback, especially the comments from billinkc:

Look at the Execution Log for the package that has run on the server. My guess is that you will eventually discover there is a permissions issue but at the moment, it could be a number of places.
Specifically, what I'd look at is the Pre/Post execute entries in the Execution report. that would indicate whether the Script task ever fired. If there's no pre/post, then look for informational events related to your Foreach Loop Containers.
If a permission issue prevents the FELC from seeing a directory, it does not error out. Instead, it fails gracefully, which trips up 99.999% of the population
If it is a validation issue (somefile.txt doesn't exist when the package begins b/c it determines what the current file is), then you will want to set the DelayValidation property to true on the Connection Manager and any tasks that reference said Connection Manager.