Baking – Flourless cake didn’t set

What you are making is a pound cake. Or maybe a sponge cake, depends on the order of mixing the ingredients. Of the two, the sponge cake is somewhat lighter, so if you have been making pound cake (cream butter and sugar, add eggs, then flour), you can switch to sponge (foam eggs and sugar, add flour and fat). But both are hearty homemade cakes, they are supposed to be dense.

The iced cakes you get in a bakery are much lighter, because they use a different type of layer. Most types of cake are lighter than pound and sponge cakes. Try making genoise, chiffon, or angel food cake layers, they are all light. And yes, using cake flour will help too, or replacing part of the flour in the recipe with pure starch (usually cornstarch or potato starch, you can also use wheat starch if you can get it. Tapioca or arrowroot aren't that well suited). For a real "pro" taste, also throw in some lecithine, it makes the mouthfeel softer and smoother - but if you don't feel like hunting around for lecithine, don't bother, a good homemade cake doesn't really need it, I only mentioned it because you asked about "cake made by a pro".

An analogue thermometer isn't too accurate. Besides, it sounds like you measured the oven temperature, not the tart filling. So I suspect that it was the heat after all.

A custard with both eggs and starch needs to be thoroughly cooked. The reason is that yolks contain an enzyme which liquidifies starch. It doesn't happen outright, but will happen while your tart is cooling. The only way to prevent it is to heat the mix high enough so the enzyme is deactivated.

To quote On Food And Cooking,

However, the survival of yolk amylase can spell disaster in the fillings for American cream pies, which are often made in the fashion of a bouillie rather than a pastry cream, and are held for hours or days before serving, enough time for a perfect cream pie to disintegrate into a soupy mess. No matter what a recipe may say, always be sure that the egg yolks in a starch-filled pie filling are heated all the way to the boil.