SSIS - sp logs counters in debug mode but not from agent job
What this appeared to boil down to was the TargetServerVersion property for the project. What I hadn't realised was, where SSDT had been upgraded, the TargetServerVersion defaults to 'SQL Server 2017', whereas my target instance is 2016.

I originally tried to "downgrade" the target server version in the original project, but was still getting the same problem. I was also getting some informational messages about the script task in the child package, saying that something hadn't gone as planned when "downgrading". I eventually recreated the entire project with the TargetServerVersion set to 2016 from the beginning. This has done the trick, as all is working as it should.

The only other thing I changed was that I deployed the project using SQL Server 2016 Deployment Wizard, instead of 2017 as before. However, I'm not too sure this would have been the problem.

Moral of the story, always check your TargetServerVersion.

I have provided this as the answer, in case other people experience such an issue. If, in future, any evidence presents itself to indicate that the above was not the cause of the issue, I will gladly change the answer.