I haven't played the game yet, so I just did a test run. It's really straightforward and I think something else is going on, as I don't think you are doing anything wrong.
To save a game, I brought up the menu, clicked on save, then I clicked on the clipboard, and the game was saved with the location, date, and time. To load a game, I brought up the menu, clicked on load, and then clicked on the save listed on the clipboard that I wanted to return to.
To quit, I brought up the menu and clicked on quit to exit the game.
I made a total of 4 saves and they all worked.
I just found this online, and I think that this might be your problem:
Q: How to run Barrow Hill on Windows Vista or Windows 7, Windows 8, Windows 10.
A: To run Barrow Hill on later Windows versions, install the game to the Documents folder or a Games folder on your root drive, and Not Program Files, to allow save game features.
Where did you install the game to?
For some reason the saved games are working for me in ProgramFilesX86, but apparently this is not always the case. Try installing the game in your Documents folder. In fact, I will probably uninstall my game and install it to Documents, just in case I run into this problem later on. Normally when I try to play an older game I install it to the Documents folder to avoid the kind of potential problem mentioned above, but in this case I just forgot!