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.
There are two likely causes of soufflé problems.
1) Egg foam. Be sure that you're whipping them to peaks (soft peaks, as I recall) when you first whip them. Then, when adding the 2 mixtures together in 3rds, be sure that you're properly folding in the whites; this is a very specific technique, and easy to over-do.
2) Oven heat. Use an oven thermometer to check your oven's temp - it's probably wrong. Put some extra mass in that oven (like a baking stone or brick) to reduce wild temperature swings during baking.
But, as in the linked questions in the comments, all soufflés fall to some degree.
Best Answer
There’s a fundamental difference between food safety and what is considered acceptable risk - the former is calculated to be virtually risk-free and statistically contain not enough bacteria etc. to potentially cause illnesses or infection, plus a bit of a safety margin. The thresholds are so that even the most sensitive and vulnerable consumers are still protected, for example young children, the elderly, pregnant women and people with weakened immune systems. These consumer groups are also especially likely to suffer or even die in case of salmonella or similar food borne diseases. This is one of the reasons why pregnant women are warned against certain foods.
But there are many recipes that are not “safe” in this strict sense. That doesn’t mean that your custard, mousse au chocolat, sauce hollandaise (all with raw egg), unpasteurized cheese (raw milk), steak tartare (raw beef) etc. will automatically make you sick. In fact, in the vast majority of all cases nothing will happen. And in most other cases, a healthy adult will suffer, but survive.
To be very clear:
Just because some cooks and some recipes work outside the boundaries of food safety guidelines doesn’t invalidate these guidelines.
The recipe in the link is not safe by official standards if made with raw, unpasteurized eggs.
The temperature you found is correct without additional time constraints, there may be ways to produce a safe custard if you can hold the mixture for a certain time at a given temperature (bacteria destruction is a function of temperature and time to be exact, food safety guidelines often give a simpler version using a temperature where the holding time is near zero, so “instant” killing of bacteria), but that would be beyond the scope of the question at hand, suffice to say that this constellation is nowhere near that range.
So if you want to make the custard recipe in your link, make an informed decision, considering your personal health status and your individual risk tolerance. I personally would never serve this kind recipe for the groups listed above or for a potluck where I can’t be sure who will participate. Especially for eggs, food safety may depend on locale, and in some areas pasteurized eggs are available in stores.