Most creme brulees require baking, however after a little research I did find a recipe in "On Cooking" (Sarah Labensky/Michael Hause) that came from Chef Vincent Guerithault of Vincent on Camelback in Phoenix, AZ and his was similar in that it was not baked.
First, just making creme anglaise with heavy cream isn't going to do anything to let it set up into a firm custard. More egg yolks or starch would be needed.
Supposing that this really does work and it was something you perhaps did, my guess would be that it was either mixed too much (breaking down the proteins trying to link together) or too vigorously (incorporating air which weakened the protein links). In your description you say you "beat it". Did you beat it or stir it? It should be stirred back and forth zig-zagging across the bottom of the pan with a wooden spoon or heat-safe rubber spatula to keep from whipping air into it.
Time, temperature, and eggs/dairy ratio are going to be the main issues in getting custards to set.
Egg proteins begin to set at 160 degrees but curdle at 180 so there's very little "wiggle" room temperature wise.
According to Shirley Corriher's "Cookwise": 2 egg yolks will just barely thicken 1 cup of milk or cream. Her Creme Anglaise recipe uses 5 egg yolks to 1 cup of milk and 1/2 cup heavy cream which is more yolks and less liquid than Julia's and this isn't intended to set up. 1 teaspoon of starch isn't going to provide the thickening power that is needed, it's there to keep the yolks from curdling as easily.
The recipe I use and many others I've referenced (including Chef Vincent's), use a ratio of about 6-7 yolks per cup of cream.
Also, if using a starch, you need to nearly bring the custard mixture to a boil (as is common in puddings and cream pie fillings) otherwise an enzyme in egg yolks known as alpha-amylase will eat away at the starch bonds and break them down into a watery mess.
Chef Vincent's does not use any starch.
If you want to use that recipe, I would increase it to 10 egg yolks. After the hot cream is tempered into the egg yolks then return to the heat and cook, stirring constantly, until very thick but do not let it boil. Remove it from the heat and strain into a clean metal bowl and chill over an ice bath to cool quickly. Once cool, spoon into your desired serving dish or a cookie cup and caramelize the top with sugar.
Best Answer
What went wrong is that you mixed up your terms and made the wrong recipe.
There are many custards. The common thing is that they are all a liquid thickened by mixing it with egg yolks and cooking them until they are just right. But from there on, you get countless variations, and they can have different names.
The name "crème anglaise" is used for a liquid custard. It is somewhat thickened (not as liquid as pure cream), and is used as a kind of sweet sauce, poured over pastries. A typical use would be to eat it with an apple strudel.
Yours turned out more liquid than in the original recipe because you had 40 g less fat and 40 g too much liquid. But even if you had not done that, the recipe is still expected to stay runny.
If you want to fill a tart, I suggest that you use a different custard, for example a crème pâtissière. It is a cooked custard (not baked like a flan) and sets as a spreadable/pipeable mass. It will use the same ingredients as a crème anglaise, but in different proportions, so the taste will be very similar.
The "don't do substitutes" thing still applies. In baking, the amount of fat and sugar in recipes is what creates the final texture, and cannot be changed due to dietary considerations. If you try, you will end up with food which is too dry, or too runny, or in some other way different from the original.