Part of the problem I think is for people using a browser on a PC the screen is very wide compared to a phone. On my screen the bright Reply and Discard buttons occupy 5.5 cm on the right of the screen, there is then a gap of over half the screen to the tiny (1.5 cm wide) private button.
Incidentally the same screen has a Previous Topic button which is greyed out as I'm on the first topic (though a lighter grey than the Private button).
If the Private button were closer to the Reply, for example underneath, it would be easier to notice and to realise it relates to the reply button.
Another suggestion. Could "Reply to Sender" appear as an option in the More menu on the message view screen? That is where I first looked for the functionality, and it would be sufficiently obscure people would not click it by accident. Just to take you to the reply screen with Private enabled.