Is this acceptable being monowhite or should it be saved for a set with multicolor so it could be ?

I would say this should really be because Propaganda had a similar effect, White is definitely not the color of discard, and blue has discard as well (Amnesia, Rise // Fall, Dismal Failure).

The Propaganda effect (along with other forms of "taxing") has been moved to white now. Recent versions of that effect are all white: Ghostly Prison, Norn's Annex, Windborn Muse, and so on. It'd be out of pie for this effect to be blue.

In fact, blue doesn't work for the second half either, because Dismal Failure is from Planar Chaos which is basically disqualified from being used as a precedent for the colour pie, Amnesia is incredibly old from before the colour pie was established, and the blue part of Rise // Fall isn't the discardy part...

White can occasionally dabble in discard. But it's normally only as part of a symmetrical balancing effect (Restore Balance) or a set theme (Patrol Hound, Trained Pronghorn). This would probably be best as a enchantment - you're right.

I guess I'll have to save probation for later.

Innistrad and New Phyrexia did both give us examples of blue discard, in Frightful Delusion and Mindculling, respectively.

Tap any number of 'untapped' permanents you control? Because otherwise, this card gains you life equal to the number of permanents you control for every card you draw. You can choose to tap tapped permanents, it just doesn't do anything... but there's a good chance it fills the requirement of "if you do"...

Both INN and NPH had black bleed into the other colors, though. Not like Torment, but certainly inspired by it.

That's true.