Yes, calling with curl should work fine for making requests with export block descriptors. In my testing the only catches were needing to authenticate using the -u flag and that the URL must be encased in single quotes. So for your example, you would form the command:
curl -u username:password ‘http://10.0.0.53/rcgi.bin/ParamForm?AST_Param=$dtHL$ftT$st20112021_000000$et_d1$fnLog.csv’
Plus whatever other flags you might need, like an output file location.This has worked for me on my device. Note that the credentials are unencrypted. This shouldn’t be an issue if you’re making the call over VPN or your local network, but you should be aware in case you decide to do so over an otherwise unencrypted connection.
I believe using the task planner to store the files you want locally should work as well.