go back to qslim's post about line pressure ~~~~~ I don't think it's simply a question of worn friction clutches or fluid, because your forward gears work fine.
I'm not a transmission expert, but would guess is something in the valve body preventing the reverse clutches from receiving sufficient hydraulic pressure to engage for reverse
Guessing, if you "tear into it " try disassembling and cleaning the valve body in solvent
Probably some tiny bit of grit in the wrong place
Total capacity for the transmission is 5.9 quarts, normal drain and fill 2.5 quarts, overnight drain and fill 3.5 quarts (apx.)
Try to get the right mind set. IMHO there's a lot more refinement in a Toyota than a Chevy. Just check wiring under the dash, for example. Think of it as if you're disassembling a Japanese camera - high level of precision, and every litttle thing has to be right to avoid problems later.
As you probably already know, intense cleanliness is required for transmission work. For example, don't use a rag to dry components, spray or soak in solvent and air dry or carefully blow dry with compressed air to maintain antiseptic cleanliness.
Could be that's what started your whole problem. Perhaps someone changed the transmission pan, wiped it down with a rag, and a bit of thread got jammed in the wrong check valve in the valve body.