If I had this requirement I would just generate a file of specific size, place it on one server and on the other I would have a shell script running via cron and measure the time it took to download the file.
It seems like a relatively simple problem.
BTW are you sure you want to test download speed and not latency? I think some routers might have the later built in.
Three streaming (like pointed in the other comment) was my initial reaction too, but indeed at the time https for streaming would be very rare.
Another possibility is to realize that openssl isn’t just for communication, but also has implementation of cryptographic algorithms.
Perhaps openssl was used for validation of licensing key? For example they could sign the license with their private key and WinAmp could verify it’s authenticity with its public key.