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.

Any reason I should go with the master rebuild kit you linked over other ones available that are much cheaper, but still with the steel plates? I found this kit on ebay which includes pistons and a "steel module" for $211.25:

4T60E Super Overhaul Rebuild Kit 1991 - 1999 | eBay

In other news, I've found a GM Technician'* Guide for the 4T60E. It has pretty pictures like this one:



Check it out:

https://www.dropbox.com/*/h93gltsnro...6615-4T60E.pdf

Dead on. I finally nutted up and dumped in the same amount of new fluid that I drained out, and the thing works fine now. I imagine it still needs to be rebuilt, and I'm going to baby the thing and drive short distances for a little while, but at least I have something to get groceries with until the rebuild kits come in.

As long as the kit has the steel plates I am pretty sure it does not mater which you go with, but hopefully one of the guys that know transmissions chimes in, as I could be wrong.

Glad to hear after the fluid change you are able to still drive for now, and that looks like a good pdf full of info on the transmission, I do not know if that is the same guide a re-builder uses though.
I do know that I hope my transmission holds up till I can afford to have it rebuilt, I am goods with mechanical things, but after looking at a few of those pics I am not so sure I should try to do it myself, not without a little help at least, if you do your own I hope the rebuild goes well for you!