Whether you rinse the rice has more to do with the rice than the recipe. If we’re speaking entirely hypothetically, the decision to rinse or not rinse depends on whether you want the starch in the rice (as in this case, to thicken a pudding) or you want the rice not to glop together as much (as in the case of sushi rice).
But in the real world, what you do to your rice has more to do with what the manufacturer did (or didn’t) do to it. Many brands of rice are enriched or otherwise treated, have normalized starch levels, and rinsing them provides no benefit. On the other hand, if you’re the sort of person who likes to buy a large burlap sack of rice from a store where they’re butchering animals in the back room, for the love of all things holy wash your rice. If you feel like you need more starch, grab one of the many, many starch-like things in your kitchen and add it later.
From a box sold in Canby, Oregon.
Old Fashioned Rice Pudding
1/3 c. unrinsed rice
1/2 tsp. salt
1/3 c. sugar
4 cups milk
1 c. raisins
2 Tbsp. butter
Combine rice, salt and sugar in shallow 8-inch square pan. Add scalded milk and raisins. Stir.
Dot with butter. Bake in slow oven, 300 deg. for 1-1/2 hours. Stir occasionally.