Can a private pilot placard an inoperative instrument?
The answer to your question is probably. The rule governing this is


  §91.213   Inoperative instruments and equipment. (a) Except as provided in paragraph (d) of this section, no person may take off an
  aircraft with inoperative instruments or equipment installed unless
  the following conditions are met:


and if you skip past the MEL section (which, as you noted, doesn’t apply to most small general aviation aircraft) you see the conditions that must be met.


  (3) The inoperative instruments and equipment are—
  
  (i) Removed from the aircraft, the cockpit control placarded, and the
  maintenance recorded in accordance with §43.9 of this chapter; or
  
  (ii) Deactivated and placarded “Inoperative.” If deactivation of the
  inoperative instrument or equipment involves maintenance, it must be
  accomplished and recorded in accordance with part 43 of this chapter;
  and
  
  (4) A determination is made by a pilot, who is certificated and
  appropriately rated under part 61 of this chapter, or by a person, who
  is certificated and appropriately rated to perform maintenance on the
  aircraft, that the inoperative instrument or equipment does not
  constitute a hazard to the aircraft.


Suppose your landing light burns out. A landing light is not required unless you are flying for hire, so you can put a piece of tape under the switch and label it Inoperative and you are good to go. 

If your transponder fails, turn it off and mark it Inoperative. If you are going to fly in airspace that requires a transponder, you have to jump through some hoops per §91.215, but you are still good.

If there is something that requires removal or disabling—and it is not something you are allowed to do in Part 43—then you need an A&amp;P to remove or disable it and make a logbook entry. I’m having a hard time thinking of something that would need disabled and yet you could still legally fly the airplane. The best I can come up with right now is a stuck push-to-talk button. That’s not listed in Part 43 as something that you can perform maintenance on, so an A&amp;P would have to disable it and make a logbook entry.

By the way, you can fly forever with inoperative equipment. There is a LORAN in plane I fly that has been marked inop since they decommissioned LORAN. When I put a new radio in my plane 10 years ago, I kept the old one in case something happened to the new one, and marked the nav portion inop.