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.
No, it doesn't have to thicken, at least not in the way custards like creme caramel thicken. "Coat the back of the spoon" is just a misnomer, a term known to experienced cooks which confuses everybody who learns to cook on their own.
For "Coat the back of the spoon", read the comment riotburn left at the question What is the correct consistency of a cream soup?. It says: "The best way to describe 'coat the back of a spoon' is where if you dipped the spoon into the soup, take it out and holder over the pot, you'll find thin film (<1mm thick)."
For the ice cream, heat to between 80 and 85°C. I've explained it somewhere else, but in short, this gives you enough thickening without overcooking. If it is really gelato and not French style, it should be thickened with starch anyway. Also, watch videos on the Internet for how to make ice cream, they show the correct consistency. I think Allrecipes had a good one, but their search is so strange, I can't find it and don't know if they have it or not.
Best Answer
The egg yolks can thicken the liquid when you heat it, like in a standard egg custard (the home made version, not the one made from custard powder). But this is a fairly tricky procedure: if you heat it too much the mixture will separate. So best done on a double boiler ('bain marie').
And in addition, you have the whipped cream and the gelatine, which will help thicken the mixture on cooling*. I suspect that there's also a prescribed time in the refrigirator...
(*: as I've been taught, the swollen gelatine is first dissolved in a small amount of liquid, and the whipped cream is mixed with the rest when the gelatine just starts to set)