How does fat affect the precipitation of proteins?
Casein is the main protein in milk. When you perform the polymerization experiment, the result is an "aggregation" of said protein, which gives the product a solid consistence.

Products such as butter and cream are made with the fats from milk (not the proteins!), hence they are almost casein-free. Of course, being them almost without proteins, the polymerization doesn't occurr.

If you look up the nutrition facts of milk and heavy cream, you see that proteins constitute the 20% of calories of milk, and only the 2% of heavy cream.