The fluid not smelling burnt is a good thing, but those plastic bits in the filter may be that 4th clutch thrust washer, here is a pic attached of mine when I removed it from the filter.
The goo is normal, if you did not find any big bits of metal, or metal shavings, then that is a good thing.
I am not 100% sure, but that washer should be in the master rebuild kit, I do not know a lot about transmissions, but am trying to learn.
I do know if you get a rebuild kit you want one with steel plates, I found a place with a good rebuild kit, they also sell the rebuild manual, and a transgo shift kit that many use when they rebuild their transmission.
I may have to try to rebuild mine some day in the future, I am not looking forward to it because it seems like a lot of tedious work.
Oregon Performance Transmission - Rebuild Parts for 700R4 4L60E 4L80E E4OD 4R100 47RE 48RE & More GM 4T60E Automatic Transmission Transaxle Master Rebuild Kit Build Olds Pontiac Chevrolet 4T60-E
When you get it back together, and the fluid topped off, hopefully it runs a bit better than it was, if so I'd do as I am, baby the thing, and change the fluid regularly, inspecting the filter as you have each time.
Forgot to mention, if it does improve the symptom you were having, then when you come due a fluid change again, you may want to have somewhere like Jiffy-Lube do a Transmission Fluid Exchange, I do not know exactly how they do it, but they get all the old fluid out of there, including the torque converter and transmission cooler.
You may be able to do this yourself, but I am unsure of what you would need to do to evacuate all of the fluid out of the transmission.