Sugarless gum makes my mouth feel somewhat clean after I eat. Does this type of gum still leave a film on teeth that should be toothpaste brushed off?

Yes, it leaves. Chewing sugarless gum can HELP prevent tooth decay and gum disease. It causes more saliva production, which helps neutralize and rinse away some of the acid that forms in your mouth when you break down food. However, even sugarless gum, is not a substitute for a regular oral hygiene.
No film. The Xylitol actually helps dissolve and remove the biofilm layer, making your teeth feel squeaky clean.