The SSDs are likely not the issue, assuming they are ok and not too worn out.
I remember having something similar on a system here, but that around 2006(?) with a 2-core AMD Athlon.
Your best bet would be to systematically eliminate components.
- How do things change if you perform a local file copy? (Eliminates the network)
- What about RAM and CPU utilization during the process?
- What about using a different protocol (NFS)?