Moving hard drives from one Mac to another
To save any potential issues with firmware changes over the years, make sure the new Mac is running the same or newer macOS as the old one was.

Mojave, for instance installed a firmware update to many machines at the time. I'm not certain this would cause any real issues if it wasn't present, but it would save any uncertainty.

Once you've done that, then yes - as the Mini Server is pre-T2 security - you should be able to simply swap out the drives with no ill-effects whatsoever.

if you have an external UDB dock that would take your system HD &amp; the new machine's 750GB SSD, I'd consider using something like Carbon Copy Cloner to clone the HD to SSD, for the added speed.