I use a different kind of cage liner, fleece holds the smell to much ( even when washed ) and isn't as easy as shaking off, everyyyyyything sticks to fleece.
Will it work yes, it's just not as easy as the liners I use. I use Jana's tray liners for all my cages
http://gr8pots.webs.com/
they are custom made to order to fit any size tray. Jana is super nice and a pleasure to work with. The liners themselves are well made, do help with the smell, are super easy to clean, and will last for years. Everything DOES shake off easily and they can be machine wash and dry.
I wash mine outside with a hose, It took 4 minutes to clean 3 tray liners.
Now it takes be about 10 minutes, I squirt some dish soap on them and suds them all up then rinse well still less time than no tray liners and they look great in the cage.
Cage sets You don't have to use allll the pieces in a big cage set, you can set some pieces aside for wash day. I'd suggest washing half of what is in the cage n leaving some dirty pieces with their scent on it in the cage. If you wash everything all at once they will work harder to stink up all the clean stuff. The more fleece you have in the cage the more you may notice the smell.
Personally I only use a fleece pouch and maybe a bridge in the cage, the smell is far less than with a cage full of fleece. Big cage sets are more or the humans sake I think to make the cage look more appealing.
cage cleaner products I'm not a fan of these, especially no rinse solutions. They build up on over time and will leave a residue behind even if you can't see. They have also been known to cause allergic reactions and skin irritation for some gliders. Dawn dish soap and hot water is sufficient to clean cages, vinegar is ok if it's not to strong, steam cleaners are awesome so are are pressure washers.
Don't make the mistake of over cleaning their cage. The more you clean, the more it can make them mark it all back up and the more stinky it will actually be. Gliders have scent glands and like to use them. Avoid strong smelling laundry soaps and cage cleaning soaps. The more you try to mask it the stronger it will be.
Covering cage Covering the cage is fine, but I wouldn't use anything plastic or vinyl ( use those behind the cage on the wall not directly on the cage ) , the cage needs air circulation. You can use a breathable fabric like fleece to prevent drafts n cover the top, back, and sides but they don't need shielded from light or covered entirely like birds do.
Good luck and congrats