I should have mentioned that I've already cut the shelves and painted them black along with the inside areas of the dresser...it looks very slick. When the components are in there it just blends nicely together. I was thinking the same thing you were about the face needing to pull out and then down...hinge work is something I'm not well versed in at all.
Yes. I was thinking something along the lines of the attached pic... If the slide needed to be limited, then 2 slots could be cut in the shelf with a pin mounted in the original drawer slide under it. If more support needed to be added, then stays from the sides. The hinges mounted like thiis would limit travel to the shelf and shelf extension butting against each other at 90 degrees. With a slide limit put in, the drawer face would also limit the opening.
Challenge I see with not coming forward before pivoting, is that the drawer fronts are "inset" style. If there isn't enough room for it to pivot in place, then you might at least have to use a round over on the top rear edge to radius it for clearance. If you also had to radius the front bottom edge, then visually, I don't think that works out.
If it did have clearance to pivot in place, another way would be to hinge the drawer front to the face below it, with the shelf mounted static. Doing this way would have the pivot point at the lower front edge. Visually, with that, there will be a gap between that lowered drawer front and the shelf. Of course if you did that, then you would need limited motion hinges and/or stays.
Another reason in design I was thinking pull out shelves is "home theater system"... Electronic components- lots of cables that you have to plug in from the rear and occasional dusting. Do you have plans for a fan(s)? A cheap alternative is a small desktop style HEPA filtered air cleaner. Low cost. Small in size. Promotes air movement (for the cooling of your components), while filtering the air... Electronic components like that are dust magnets.